# Solving Linear Programming Problem with Fuzzy Right Hand Sides: A Penalty Method

Save this PDF as:

Size: px
Start display at page:

Download "Solving Linear Programming Problem with Fuzzy Right Hand Sides: A Penalty Method"

## Transcription

1 Solving Linear Programming Problem with Fuzzy Right Hand Sides: A Penalty Method S.H. Nasseri and Z. Alizadeh Department of Mathematics, University of Mazandaran, Babolsar, Iran Abstract Linear programming problems with trapezoidal fuzzy variables (FVLP) have recently attracted some interest. Some methods have been developed for solving these problems. Fuzzy primal and dual simplex algorithms have been recently proposed to solve these problems. These methods have been developed with the assumption that an initial Basic Feasible Solution (BFS) is at hand. In many cases, finding such a BFS is not straightforward and some works may be needed to get the simplex algorithm started. In this paper, we propose a penalty method to solve FVLP problems in which the BFS is not readily available. Introduction Many application problems, modeled as mathematical programming problems, may be formulated with uncertainty. Fuzzy programming approach is useful to treat a problem under Corresponding author. (Hadi Nasseri) 38

2 uncertainty. The concept of decision making in fuzzy environment was first proposed by Bellman and Zadeh [2]. Several kinds of fuzzy linear programming problems have appeared in the literature and different methods have been proposed to solve such problems. The first formulation of Fuzzy Linear Programming (FLP) is proposed by Zimmermann [2]. Some methods have been developed for solving these problems by introducing and solving certain auxiliary problems. Mahdavi-Amiri et al. [0] have developed a new fuzzy primal and dual simplex algorithm for solving the FVLP problem. As they have been mentioned the fuzzy primal algorithm needs an initial basic feasible solution (see[0]). In some cases, such a solution is not straightforward. Hence in this paper we propose a fuzzy Big-M method for solving these types of problems where directly can find a basic feasible solution if there exist. Our discussion, here is outlined as follows: In next section we briefly give some necessary concepts of fuzzy set theory. In Section 3 we first define a Linear Programming with Fuzzy Variables (FVLP) problem and then formulate a real problem as FVLP problem. Furthermore,the fundamental concepts and theorems are given to complete our discussion in this section. We also introduce a Big-M method for solving FVLP problems and illustrate it with an example in Section 4. Finally, we conclude in Section 5. 2 Preliminaries We review the fundamental notations of fuzzy set theory, initiated by Bellman and Zadeh [2], to be used throughout this note and which is taken from ([5],[6],[8],[9]and[0]). We give here, some basic definitions on fuzzy numbers comparison that will be used for illustrating our approach. In this way, and for methodological reasons, in spite that the approach that is designed will work correctly for any kind of fuzzy numbers, we prefer in the following to focus on the usual case of trapezoidal fuzzy numbers. Definition. Let X be the universal set. Ã is called a fuzzy set in X if Ã is a set of ordered pairs Ã = {( x, µã (x) ) x X }, where µã (x) is the membership function of x in Ã. remark. The membership function of Ã specifies the degree of membership of element x in fuzzy set Ã (in fact, µ Ã shows the degree that x belongs to Ã). Definition 2. The α-level set of Ã is the set Ãα = { x R µã (x) α }, where α (0, ]. The lower and upper bounds of any α-level set Ãα are represented by finite numbers inf x Ã α and sup x Ã α. Definition 3. The support of a fuzzy set Ã is a set of elements in X for which µ Ã (x) is positive, that is, suppã = { x X µã (x) > 0 }. 39

3 Definition 4. A fuzzy set Ã is convex if µã (λx + ( λ) y) min { µã (x), µã (y) }, x, y Xand λ [0, ]. Definition 5. A convex fuzzy set Ã on R is a fuzzy number if the following conditions hold: (a)its membership function is piecewise continuous. (b)there exist three intervals [a, b],[b, c] and [c, d] such that µã is increasing on [a, b], equal to on [b, c], decreasing on [c, d] and aqual to 0 elsewhere. remark 2. In the above definition, we say interval [b, c] is the modal set of fuzzy number Ã. Definition 6. Let Ã = ( a L, a U, α, β ) denote the trapezoidal fuzzy number, where [ a L α, a U + β ] is the support of Ã and[ a L, a U] its modal set. remark 3. We denote the set of all trapezoidal fuzzy numbers by F (R). If a = a L = a U then we obtain a triangular fuzzy number, and we show it with Ã = (a, α, β). 2. Arithmetic on fuzzy numbers Let ã = ( a L, a U, α, β ) and b = ( b L, b U, γ, θ ) be two trapezoidal fuzzy numbers. Define x > 0, x R; x.ã = ( xa L, xa U, xα, xβ ), x < 0, x R; x.ã = ( xa U, xa L, xβ, xα ), ã + b = ( a L + b L, a U + b U, α + γ, β + θ ), ã b = ( a L b U, a U b L, α + θ, β + γ ). 2.2 Ranking function An effective approach for ordering the elements of F (R) is also to define a ranking function R : F (R) R which maps each fuzzy number into the real line, where a natural order exists. We define orders on F (R) by: ã b ) if and only if R (ã) R ( b ã b ) if and only if R (ã) R ( b ã = b ) if and only if R (ã) = R ( b where ã and b are in F (R). Also we write ã b if and only if b ã. We restrict our attention to linear ranking functions, that is, a ranking function R such that 320

4 ( R kã + b ) ) = kr (ã) + R ( b for any ã and b belonging to F (R) and any k R. We consider the linear ranking functions on F (R) as: R (ã) = c L a L + c U a U + c α α + c β β, where ã = ( a L, a U, α, β ), and c L, c U, c α, c β are constants, at least one of which is nonzero. A special version of the above linear ranking function was first proposed by Yager[] as follows: which reduces to R (ã) = 2 0 (inf ã λ + sup ã λ ) dλ R (ã) = al +a U + (β α). 2 4 Then, for trapezoidal fuzzy numbers ã = ( a L, a U, α, β ) and b = ( b L, b U, γ, θ ), we have ã b if and only if a L + a U + (β α) 2 bl + b U + (θ γ). 2 3 Fuzzy linear programme 3. Definition of model Here we first define a general form of the problem which will be discussed in this paper. Consider a Linear Programming with Fuzzy Variables (FVLP) as follows: min z c x () s.t. A x b x 0 where c R n, x (F (R)) n, A R m n, b (F (R)) m and b 0. Note that an FVLP problem is a linear programming problem in fuzzy environment in which the decision making variables and right-hand sides are fuzzy numbers. 32

5 3.2 Formulation of a problem An alloy producer, produces 2 types of alloys P and P 2. These alloys consists of Zinc and Tin(F, F 2 ). The tableau below shows the amount of F and F 2 used in per unit of alloys. The average cost of per kilogram of P and P 2 are 0 and 6 cents, respectively. The minimum daily supplyes metals F and F 2 are approximately 49 and 45 units, respectively. The producer wants to know in order to minimize the cost of Zinc and Tin how many kilograms of alloys P and P 2 he must produce daily? P roducts Metal P P 2 F 6 F Now if we assume that the fuzzy variables x and x 2 be the daily supplyes metals F and F 2 respectively, then this problem can be formulated as follows: min z 0 x + 6 x 2 s.t. x + 6 x 2 (46, 52, 2, 2) 4 x + 2 x 2 (42, 48, 4, 4) x, x Fuzzy basic feasible solution First of all, we briefly describe Fuzzy Basic Feasible Solution (FBFS) for the FVLP problem as established by Mahdavi-Amiri and Nasseri [8]. Consider the FVLP problem and A = [a ij ] m n. Assume rank(a) = m. Partition A as [B N] where B, m m, is nonsingular. It is obvious that rank(b) = m. Let y j be the solution to By = a j. It is apparent that the basic solution x B ( x B,..., x Bm ) T B b, xn 0 is a solution of A x = b. In fact, x = ( x T T B N) xt. If xb 0, then the basic solution is feasible and the corresponding fuzzy objective value is: z c B x B, where c B = (c B,..., c Bm ). Now, corresponding to every nonbasic variable x j, j n, j B i, i =,..., m, define z j = c B y j = c B B a j. The following result concerns the non-degenerated problems, where every fuzzy basic variable corresponding to every basis B is positive. Observe that for any basic index j = B i, i m, we have B a j = e i where e i = (0,..., 0,, 0,..., 0) T is the ith unit vector, since Be i = [a B,..., a Bi,..., a Bm ] e i = a Bi = a j, and so we have: 322

6 z j c j = c B B a j c j = c B e i c j = c Bi c j = c j c j = 0. The following theorem characterizes optimal solutions. Theorem. (optimality conditions) Assume the FVLP problem is non-degenerate. If a basic solution x B = B b, xn 0 is feasible to ()and z j c j for all j, j n then the fuzzy basic solution is a fuzzy optimal solution to (). Proof: See Mahdavi-Amiri and Nasseri [8]. Theorem 2. Let x B = B b is a fuzzy basic feasible solution of the problem max z c x such that A x b and x 0. If for any column a j in A which is not in B, the condition (z j c j ) < 0 hold and y ij > 0 for some i, j {, 2,..., m} then it is possible to obtain a new fuzzy basic feasible solution by replacing one of the columns in B by a j. Proof: See[8]. Theorem 3. If in an FVLP problem, there is a column a j in A (not in basis) for which z j c j < 0 and y ij 0, i =,..., m, then the FVLP problem is unbounded. Proof: It is straight forward. We write the above FVLP problem in the following tableau format: Basis x B x N R.H.S. z 0 z N c N = c B B N c N ỹ 00 = c B B b x B I Y = B N ỹ 0 = B b Now we are going to give the concept of pivoting and change of basis. If x k enters the basis and x Br leaves the basis, then pivoting on y rk in the simplex tableau is stated as follows:. Divide row r by y rk 2. For i = 0,,..., m and i r, update the ith row by adding to it y ik times the new rth row. 4 Big-M Method The simplex algorithm was conceived by Dantzig for solving Linear Programming (LP)problems. This method starts with a Basic Feasible Solution (BFS)and moves to an improved BFS, until the optimal point is reached or else unboundedness of the objective function is verified. 323

7 In order to initialize this algorithm a BFS must be available. In many cases, finding such a BFS is not straightforward and some work may be needed to get the simplex algorithm started.to this end, there are two techniques in linear programming literature: Two-phase method and Big-M method []. But there may be some LP models for which there are not any BFSs, i.e., the model is infeasible. Both Two-phase method and Big-M method distinguish the infeasibility. Here we focus on Big-M method. Consider a generic LP model. After manipulating the constraints and introducing the required slack variables, the constraints are put in the format AX = b, X 0 where A is a m n matrix and b 0 an m vector. Considering C as cost vector, the following LP model is dealt with: min CX (P ) s.t. AX = b X 0 Furthermore, suppose that we do not have a starting BFS for simplex metod, i.e, A has no identity submatrix. In this case we shall resort to the artificial variables to get a starting BFS, and then use the simplex method itself and get rid of these artificial variables. The use of artificial variables to obtain a starting BFS was first provided by Dantzig [3, 4]. To illustrate, suppose that we change the restrictions by adding an artificial vector R leading to the system AX + R = b, (X, R) 0. This forces an identity submatrix corresponding to the artificial vector and gives an immediate BFS of the new system, namely (X = 0, R = b). Even though we now have a starting BFS and the simplex method can be applied, we have in effect changed the problem. In order to get back to our original problem, we must force these artificial variables to zero, because AX = b AX + R = b, R = 0. In Big-M method we assign a large penalty coefficient to these variables in the original objective function in such a way as to make their presence in the basis at a positive level very unattractive from the objective function point of view. More specifically, (P ) is changed to: min CX + MR P (M) s.t. AX + R = b (X, R) 0 324

8 where M is a very large positve number. The term MR can be interpreted as a penalty to be paid by any solution with R 0. Therefore the simplex method itself will try to get the artificial variables out of the basis, and then continue to find an optimal solution of the original problem. Hereafter * indicates the optimality, z j c j is the reduced cost of the jth variable, and y j = B a j, where B is the basis of the simplex method associated with the related iteration and a j is the jth column of the technological coefficients matrix (for P (M) it is [A I]). Four possible cases may arise while solving P(M): (A ) : (X, R ) is an optimal solution of P(M), in which R = 0 (A 2 ) : (X, R ) is an optimal solution of P(M), in which R 0 (B ) : z k c k = max (z j c j ) > 0, y k 0, and all artificials are equal to zero. (B 2 ) : z k c k = max (z j c j ) > 0, y k 0, and not all artificials are equal to zero. remark 4. A Two-phase method for solving FVLP problem was proposed later [7]. Here we are going to solve an FVLP problem using Big-M method. Consider the following problem: min Z = 0 x + 6 x 2 s.t. x + 6 x (46, 52, 2, 2) 4 x + 2 x 2 (42, 48, 4, 4) x, x 2 0 First of all we add the slack variables: min Z = 0 x + 6 x 2 s.t. x + 6 x x 3 = (46, 52, 2, 2) 4 x + 2 x 2 x 4 = (42, 48, 4, 4) x, x 2, x 3, x 4 0 As its seen no initial basic variable is available and hence we may use the Big-M method to solve the problem. Therefore, the problem changes to the following form: min Z = 0 x + 6 x 2 + M R + M R 2 s.t. x + 6 x x 3 + R = (46, 52, 2, 2) 325

9 4 x + 2 x 2 x 4 + R 2 = (42, 48, 4, 4) x, x 2, x 3, x 4, R, R 2 0 The problem is written in the tableau format as: Basis x x 2 x 3 x 4 R R2 R.H.S. z M M 0 R (46, 52, 2, 2) R (42, 48, 4, 4) Because R and R 2 are basic variables, so the cost row must be equal to zero. Hence the next tableau will be as following: Basis x x 2 x 3 x 4 R R2 R.H.S. z 0 + 5M 6 + 8M M M 0 0 (88M, 00M, 6M, 6M) R (46, 52, 2, 2) R (42, 48, 4, 4) It is obvious that x 2 is an entering fuzzy variable and R is a leaving fuzzy variable. Then after pivoting the next tableau is given as: Basis x x 2 x 3 x 4 R R2 R.H.S. z M 0 M 3 M 4 3 M 0 ( M 3, M 3, 2 + 4M 3, 2 + 4M 3 ) x ( 46 6, 52 6, 2 6, 2 6 ) R ( 74 3, 98 3, 4 3, 4 3 ) So x is an entering fuzzy variable and R 2 is a leaving fuzzy variable. The last tableau is given in the below. Now we can see all artificial variables leaves the basis. So we can remove their columns Basis x x 2 x 3 x 4 R R2 R.H.S. z x x ( 48, 478, 2 22 ( 68, 3 ( 74, Therefore, the fuzzy optimal solution of the FVLP problem which is obtained by the Big-M method is x = ( 74, 98, 4, ) 4, x2 = ( 68, 83, 6, 6 ) and the fuzzy optimal value of its objective function is z = ( 48, 478, 76, )

10 5 Conclusions The Big-M method is a known auxiliary technique to start the simplex algorithm. In particular, we have illustrated the mentiond method by formulating and solving a linear programming problem with fuzzy variables. 6 Acknowledgment The authors would like to thank from the anonymous referees for their valuable comments to improve the earlier version of this paper. References [] M.S. Bazarra, J.J. Jarvis and H.D. Sherali, Linear Programming and Network Flows, John Wiley and Sons, 990. [2] R.E. Bellman, L.A. Zadeh, Decision making in a fuzzy environment, Manag. Sci. 7(970) [3] G.B. Dantzig, Maximization of a linear function of variables subject to linear inequalities, T.C. Koopmans(Ed.), Activity Analysis of Production and Allocation, John Wiley and Sons, New York, 95, pp [4] G.B. Dantzig, Programming in a linear structure, Comptroller, United Air Force, Washington, D.C., February, 948. [5] D. Dubois, H. Prade, Fuzzy Sets and Systems, Theory and Applications, Academic Press, New York, 980. [6] A. Ebrahimnejad, S.H. Nasseri and S.M. Mansourzadeh, Bounded Primal Simplex Algorithm for Bounded Linear Programming with Fuzzy Cost Coefficients, International Journal of Operations Research and Information Systems, 2()January-March 20, [7] B. Khabiri, S.H. Nasseri and Z. Alizadeh, Starting fuzzy solution for the fuzzy primal simplex algorithm using a fuzzy two-phase method, The 5th International Conference on fuzzy Information and Engineering (ICFIE), Oct.4-6, 20, Chengdu, China. 327

11 [8] N. Mahdavi-Amiri and S.H. Nasseri, Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables, Fuzzy Sets and Systems 58(2007) [9] N. Mahdavi-Amiri and S.H. Nasseri, Duality in fuzzy number linear programming by use of a certain linear ranking function, Applied Mathematics and Computation 80 (2006) [0] N. Mahdavi-Amiri, S.H. Nasseri and A. Yazdani, Fuzzy primal simplex algorithms for solving fuzzy linear programming problems, Iranian Journal of Operational Research (2),(2009) [] R.R.Yager, A procedure for ordering fuzzy subsets of the unit interval, Information Sciences 24, (98)43-6. [2] H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, (978)

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### 4.6 Linear Programming duality

4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

### Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

### 3 Does the Simplex Algorithm Work?

Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

### LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right

### Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints

Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and

### 1 Solving LPs: The Simplex Algorithm of George Dantzig

Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

### 1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

### Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.

1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that

### Duality in Linear Programming

Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow

### Convex Optimization SVM s and Kernel Machines

Convex Optimization SVM s and Kernel Machines S.V.N. Vishy Vishwanathan vishy@axiom.anu.edu.au National ICT of Australia and Australian National University Thanks to Alex Smola and Stéphane Canu S.V.N.

### Special Situations in the Simplex Algorithm

Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### International Doctoral School Algorithmic Decision Theory: MCDA and MOO

International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire

### Linear Programming in Matrix Form

Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,

### Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

### Sensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS

Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and

### Solving Linear Programs

Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,

### IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2

IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3

### Solving Linear Systems, Continued and The Inverse of a Matrix

, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

### Linear Programming: Theory and Applications

Linear Programming: Theory and Applications Catherine Lewis May 11, 2008 1 Contents 1 Introduction to Linear Programming 3 1.1 What is a linear program?...................... 3 1.2 Assumptions.............................

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### Chapter 2 Solving Linear Programs

Chapter 2 Solving Linear Programs Companion slides of Applied Mathematical Programming by Bradley, Hax, and Magnanti (Addison-Wesley, 1977) prepared by José Fernando Oliveira Maria Antónia Carravilla A

### Two-Stage Stochastic Linear Programs

Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic

### OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

### Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

### Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1

Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear

### A New Method for Multi-Objective Linear Programming Models with Fuzzy Random Variables

383838383813 Journal of Uncertain Systems Vol.6, No.1, pp.38-50, 2012 Online at: www.jus.org.uk A New Method for Multi-Objective Linear Programming Models with Fuzzy Random Variables Javad Nematian * Department

### FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM

International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 34-48 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT

### Linear Programming I

Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

### Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

### A New Operation on Triangular Fuzzy Number for. Solving Fuzzy Linear Programming Problem

Applied Mathematical Sciences, Vol. 6, 2012, no. 11, 525 532 A New Operation on Triangular Fuzzy Number for Solving Fuzzy Linear Programming Problem A. Nagoor Gani PG and Research Department of Mathematics

### Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood

PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720-E October 1998 \$3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced

### Optimization under fuzzy if-then rules

Optimization under fuzzy if-then rules Christer Carlsson christer.carlsson@abo.fi Robert Fullér rfuller@abo.fi Abstract The aim of this paper is to introduce a novel statement of fuzzy mathematical programming

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Proximal mapping via network optimization

L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### 7.4 Linear Programming: The Simplex Method

7.4 Linear Programming: The Simplex Method For linear programming problems with more than two variables, the graphical method is usually impossible, so the simplex method is used. Because the simplex method

### Standard Form of a Linear Programming Problem

494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,

### Linear Programming. March 14, 2014

Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

### This exposition of linear programming

Linear Programming and the Simplex Method David Gale This exposition of linear programming and the simplex method is intended as a companion piece to the article in this issue on the life and work of George

### Simplex method summary

Simplex method summary Problem: optimize a linear objective, subject to linear constraints 1. Step 1: Convert to standard form: variables on right-hand side, positive constant on left slack variables for

### Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

### FE670 Algorithmic Trading Strategies. Stevens Institute of Technology

FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 Mean-Variance Analysis: Overview 2 Classical

### 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

### The Equivalence of Linear Programs and Zero-Sum Games

The Equivalence of Linear Programs and Zero-Sum Games Ilan Adler, IEOR Dep, UC Berkeley adler@ieor.berkeley.edu Abstract In 1951, Dantzig showed the equivalence of linear programming problems and two-person

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### What is Linear Programming?

Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

### . P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2

4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the

### Portfolio selection based on upper and lower exponential possibility distributions

European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### Chapter 6: Sensitivity Analysis

Chapter 6: Sensitivity Analysis Suppose that you have just completed a linear programming solution which will have a major impact on your company, such as determining how much to increase the overall production

### 4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

### Nonlinear Optimization: Algorithms 3: Interior-point methods

Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

### Factorization Theorems

Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

### Minimally Infeasible Set Partitioning Problems with Balanced Constraints

Minimally Infeasible Set Partitioning Problems with alanced Constraints Michele Conforti, Marco Di Summa, Giacomo Zambelli January, 2005 Revised February, 2006 Abstract We study properties of systems of

### Linear Programming Notes VII Sensitivity Analysis

Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization

### Discrete Optimization

Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

Quadratic Functions, Optimization, and Quadratic Forms Robert M. Freund February, 2004 2004 Massachusetts Institute of echnology. 1 2 1 Quadratic Optimization A quadratic optimization problem is an optimization

### Unit 1. Today I am going to discuss about Transportation problem. First question that comes in our mind is what is a transportation problem?

Unit 1 Lesson 14: Transportation Models Learning Objective : What is a Transportation Problem? How can we convert a transportation problem into a linear programming problem? How to form a Transportation

### 3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max

SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### A Direct Numerical Method for Observability Analysis

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 15, NO 2, MAY 2000 625 A Direct Numerical Method for Observability Analysis Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper presents an algebraic method

### 1. (a) Multiply by negative one to make the problem into a min: 170A 170B 172A 172B 172C Antonovics Foster Groves 80 88

Econ 172A, W2001: Final Examination, Possible Answers There were 480 possible points. The first question was worth 80 points (40,30,10); the second question was worth 60 points (10 points for the first

### Some representability and duality results for convex mixed-integer programs.

Some representability and duality results for convex mixed-integer programs. Santanu S. Dey Joint work with Diego Morán and Juan Pablo Vielma December 17, 2012. Introduction About Motivation Mixed integer

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

### 56:171 Operations Research Midterm Exam Solutions Fall 2001

56:171 Operations Research Midterm Exam Solutions Fall 2001 True/False: Indicate by "+" or "o" whether each statement is "true" or "false", respectively: o_ 1. If a primal LP constraint is slack at the

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

### 56:171. Operations Research -- Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa.

56:171 Operations Research -- Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa Homework #1 (1.) Linear Programming Model Formulation. SunCo processes

### 26 Linear Programming

The greatest flood has the soonest ebb; the sorest tempest the most sudden calm; the hottest love the coldest end; and from the deepest desire oftentimes ensues the deadliest hate. Th extremes of glory

### Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725

Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T

### Solutions Of Some Non-Linear Programming Problems BIJAN KUMAR PATEL. Master of Science in Mathematics. Prof. ANIL KUMAR

Solutions Of Some Non-Linear Programming Problems A PROJECT REPORT submitted by BIJAN KUMAR PATEL for the partial fulfilment for the award of the degree of Master of Science in Mathematics under the supervision

### EC9A0: Pre-sessional Advanced Mathematics Course

University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond

### OPTIMIZATION. Schedules. Notation. Index

Easter Term 00 Richard Weber OPTIMIZATION Contents Schedules Notation Index iii iv v Preliminaries. Linear programming............................ Optimization under constraints......................3

### Matrix Differentiation

1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### Option Pricing for a General Stock Model in Discrete Time

University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations August 2014 Option Pricing for a General Stock Model in Discrete Time Cindy Lynn Nichols University of Wisconsin-Milwaukee

### Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

### A Robust Formulation of the Uncertain Set Covering Problem

A Robust Formulation of the Uncertain Set Covering Problem Dirk Degel Pascal Lutter Chair of Management, especially Operations Research Ruhr-University Bochum Universitaetsstrasse 150, 44801 Bochum, Germany

### Optimization Modeling for Mining Engineers

Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2

### Lecture 2: August 29. Linear Programming (part I)

10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.

### 9.4 THE SIMPLEX METHOD: MINIMIZATION

SECTION 9 THE SIMPLEX METHOD: MINIMIZATION 59 The accounting firm in Exercise raises its charge for an audit to \$5 What number of audits and tax returns will bring in a maximum revenue? In the simplex

### Advanced Lecture on Mathematical Science and Information Science I. Optimization in Finance

Advanced Lecture on Mathematical Science and Information Science I Optimization in Finance Reha H. Tütüncü Visiting Associate Professor Dept. of Mathematical and Computing Sciences Tokyo Institute of Technology

### 7 Gaussian Elimination and LU Factorization

7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method

### Lecture Note on Linear Algebra 15. Dimension and Rank

Lecture Note on Linear Algebra 15. Dimension and Rank Wei-Shi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter

### Can linear programs solve NP-hard problems?

Can linear programs solve NP-hard problems? p. 1/9 Can linear programs solve NP-hard problems? Ronald de Wolf Linear programs Can linear programs solve NP-hard problems? p. 2/9 Can linear programs solve

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### 6. Cholesky factorization

6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

### Equilibrium computation: Part 1

Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Optimization in R n Introduction

Optimization in R n Introduction Rudi Pendavingh Eindhoven Technical University Optimization in R n, lecture Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4 Some optimization problems designing

### Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

### LU Factorization Method to Solve Linear Programming Problem

Website: wwwijetaecom (ISSN 2250-2459 ISO 9001:2008 Certified Journal Volume 4 Issue 4 April 2014) LU Factorization Method to Solve Linear Programming Problem S M Chinchole 1 A P Bhadane 2 12 Assistant