# Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC Coimbra

Save this PDF as:

Size: px
Start display at page:

Download "Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC Coimbra"

## Transcription

1 Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC Coimbra João Clímaco and Marta Pascoal A new method to detere unsupported non-doated solutions in multicriteria integer linear programg - a reference point approach No ISSN: Instituto de Engenharia de Sistemas e Computadores de Coimbra INESC Coimbra Rua Antero de Quental, 199; Coimbra; Portugal

2 A new method to detere unsupported non-doated solutions in multicriteria integer linear programg - a reference point approach João C. N. Clímaco (1), Marta M. B. Pascoal (1,2) (1) Instituto de Engenharia de Sistemas e Computadores Coimbra, Universidade de Coimbra Rua Antero de Quental, 199, Coimbra, Portugal (2) Departamento de Matemática da FCTUC, Apartado 3008, EC Santa Cruz, Coimbra, Portugal February 18, 2012 Abstract: In this paper we introduce a method for finding both supported and unsupported non-doated solutions of a multicriteria integer linear program (MCILP). This consists of two phases, in the first a weighted-sum method is used that finds only the supported solutions of the problem, in the second the Chebyshev distance to a reference point is imized in order to sweep the region between any pair of adjacent supported solutions and thus find the unsupported solutions. Computational experiments are discussed. Keywords: Multicriteria problems, Supported and unsupported non-doated solutions, Two-phase method, Reference point, Chebyshev metrics. 1 Introduction Let us consider the multicriteria integer linear program (MCILP) with k objective functions, n variables and m constraints: (x) = c T 1 x f 2 (x) = c T 2 x f k (x) = c T k x s. t. Ax b x 0 x ZZ n where A IR m n, b IR m and c r IR n, for any r = 1,..., k. We assume that S = {x ZZ n : Ax b, x 0} is nonempty and bounded. For easiness of presentation we consider that all variables are integer, however the method presented in the following still applies to multicriteria mixed integer linear problems. A solution x S of problem (1) is efficient if and only if there is no other x S such that f(x ) f(x) and f(x ) f(x). A criterion vector f(x) is non-doated if and only if x is efficient. We focus on the deteration of non-doated solutions, which means that if there are alternative efficient solutions with the same image then only one of them is calculated. Despite these two standard definitions, in the remaining of the paper we might use the terms efficient solution and non-doated solution interchangeably. All the non-doated solutions of a MCILP form the problem s Pareto frontier. We distinguish two types of non-doated solutions, 2 (1)

3 supported non-doated solutions, which are non-doated solutions that are optimal solutions of a (single-criterion) weighted sum problem (WSP) x S { k } w r f r (x) : w r 0, r = 1, 2,..., k ; (2) r=1 the remaining non-doated solutions are called unsupported solutions, and they cannot be obtained as solutions of a WSP. While on the one hand supported non-doated solutions can be found by means of solving a single-criterion WSP (if there exists an efficient algorithm for the problem), on the other finding unsupported solutions is usually a more difficult task and this type of solutions remain to characterize. Because of these differences the calculation of the two types of solutions is often done in two phases, first the supported solutions are computed, afterwards the search for unsupported solutions continues by looking for solutions the images of which lie within the duality gaps formed by two adjacent supported solutions. For further details see [1, 5, 7]. There has been some work on methods for the first phase, however, to our knowledge, the problem of finding unsupported solutions has merited little attention in the literature. The search of solutions within duality gaps has been investigated since Current et al. [4]. Two main approaches have been proposed, namely using side constraints to obtain a sequence of easier problems, and using an algorithm able to rank the K best solutions of the problem to sweep that area. Whenever ranking solutions can be made fast the latter approach has revealed to be quite efficient. The bicriteria shortest paths problem or the bicriteria spanning trees problem are two of those cases, in the first case the search can be done quite easily, [3], the second case is harder but still manageble for problems that are not too big, [6]. However, in the general case it can even be more difficult to rank solutions, and therefore more difficult to perform the search. The method presented in the following has a broader application, and the empirical experiments show it can be used in practice. In the next section the second phase method for computing the Pareto frontier of a MCILP is described. A weighted-sum method (WSM) for finding the supported solutions is outlined, and then an algorithm for computing all the unsupported solutions within two adjacent supported solutions is proposed. At the end of the section an example of the application of the algorithm is given, whereas Section 3 is devoted to computational results. 2 Algorithmic approach Figure 1 illustrates the different types of solutions of a MCILP. Among the non-doated solutions we distinguish the supported and the unsupported. In the present section we describe a method for computing the Pareto frontier of a MCILP. The first part of the method proposed in the following is responsible for the supported solutions calculation, while the remaining non-doated solutions, 3

4 f 2 s 1 s 3 doated solution non-doated unsupported solution non-doated supported solution s 2 Figure 1: MCILP solutions i.e., the unsupported, are calculated during the second part. In order to simplify the presentation we only consider the bicriteria case, with k = 2, however the results can be easily extended to a general k. The WSM calculates the non-doated solutions by solving a sequence of WSPs and updating their parameters according to the solutions found. Descriptions of the WSM can be found in [3] or [7]. The method works by starting with the lexicographic best solution with respect to each criteria, and then taking pairs of consecutive solutions, s 1, s 2, and, for each, calculating a new one by considering a single criteria WSP. The goal is to detere whether there exist further supported non-doated solutions within the triangle f(s 1 )c f(s 2 ), where c = (c 1, c 2 ) and c i = f i (s i ), i = 1, 2 Figure 2.(a). The new solution, s 3 depicted in Figure 2.(b), is the best regarding the imization of the objective function w 1 (s) + w 2 f 2 (s), with w 1 = f 2 (s 1 ) f 2 (s 2 ), w 2 = (s 2 ) (s 1 ), and because the solutions are restricted to the mentioned triangle the problem can be formulated as w 1 (s) + w 2 f 2 (s) s. t. s S (3) It should be added that the weights w 1, w 2 can also be normalized. This approach is inspired in the noninferior set estimation (NISE) method [2]. The process is illustrated in Figure 2. A summary of the whole phase 1 method is included in Algorithm 1. Algorithm 1 (Supported non-doated solutions deteration). // (s 1, s 2 ) identifies a pair of consecutive solutions // L is a set that stores the calculated non-doated solutions // X is an auxiliary set that stores the calculated pairs of solutions s 1 lexicographic best solution with respect to (, f 2 ) s 2 lexicographic best solution with respect to (f 2, ) L {s 1, s 2 } X {(s 1, s 2 )} While X Do (s 1, s 2 ) element in X; X X {(s 1, s 2 )} w 1 f 2 (s 1 ) f 2 (s 2 ); w 2 (s 2 ) (s 1 ) 4

5 f 2 s 1 f 2 s 1 s 3 c 2 s 2 c 2 s 2 c 1 (a) c 1 (b) Figure 2: Phase 1, weighted-sum method: (a) initial supported solutions; (b) supported solution obtained from the previous pair of solutions s best solution to (3) If s is defined Then L L {s} If (s) (s 2 ) Then X X {(s 1, s)} If f 2 (s) f 2 (s 1 ) Then X X {(s, s 2 )} EndIf EndWhile Notice that Algorithm 1 calculates at most one new supported solution given a pair of consecutive supported solutions. This process can also be adapted in order to privilege or limit the search to a certain region, for instance in an interactive method. f 2 ĉ 2 s 1 s 3 f 2 ĉ 2 A s 1 C s 3 D c 2 s 2 c 2 B s 2 c 1 (a) ĉ 1 c 1 (b) ĉ 1 Figure 3: Phase 2, computing unsupported solutions: (a) initial search area (shaded); (b) areas to scan (shaded) after the unsupported solution s 3 is calculated in search for possible non-doated solutions The goal of the second part of the method is to search for other solutions lying within the duality gaps formed by pairs of adjacent supported solutions obtained by Algorithm 1. Non-supported solutions are not the optimal solution of any WSP, and thus Algorithm 1 is not able to find the whole Pareto-frontier. However, Theorem 1 states that unsupported non-doated solutions are solutions closest to the ideal point, the point defined by the best value of every criteria, with respect to a certain weighted Chebyshev distance [8]. Theorem 1. Let s i be the optimal solution with respect to f i, and c i = f i(s i ) represent the optimal value for each criteria, i = 1, 2. If s is a non-doated solution of (1) then it is the optimal 5

6 solution of for some w 1, w 2 0 with w 1 + w 2 = 1. max{w 1 (s) c 1, w 2 f 2 (s) c 2 } s. t. s S A consequence of this result is that any unsupported solution can be found by using a Chebyshev metric with adequate parameters. Taking now s 1 and s 2 as two adjacent supported solutions, we want to apply Theorem 1 in order to search for unsupported solutions within the duality gap given by the shaded area in Figure 3.(a). This is equivalent to look for non-doated solutions within the rectangle defined by the points c, f(s 1 ), ( (s 2 ), f 2 (s 1 )), f(s 2 ), where c = (c 1, c 2 ) and c i = f i(s i ), i = 1, 2. If the weights are w 1 = f 2 (s 1 ) f 2 (s 2 ), w 2 = (s 2 ) (s 1 ), and the reference point is c, given above, then solving the problem T (s) = max{w 1 (s) c 1, w 2 f 2 (s) c 2 } s. t. s S (4) with the weighted Chebyshev metric, finds the best solution with respect to an objective function the level lines of which are proportional to the latter rectangle taking c as the reference point, as illustrated in Figure 3.(a) and shown in Proposition 1. Proposition 1. Given s 1, s 2 a pair of adjacent supported non-doated solutions of (3), the optimal solution of (4) is the best solution with respect to lines proportional to the rectangle with vertices c, f(s 1 ), ( (s 2 ), f 2 (s 1 )) and f(s 2 ). Proof. Let s 1, s 2 be a pair of solutions of (3), and s be the optimal solution in S regarding T. Assume there is another solution s S such that { c 1 (s) < (s ) c 2 f 2(s) < f 2 (s ) Then, { 0 f1 (s) c 1 < (s ) c 1 and therefore 0 f 2 (s) c 2 < f 2(s ) c 2 { w1 (s) c 1 < w 1 (s ) c 1 w 2 f 2 (s) c 2 < w 2 f 2 (s ) c 2 max{w 1 (s) c 1, w 2 f 2 (s) c 2 } < max{w 1 (s ) c 1, w 2 f 2 (s ) c 2 }, which implies that s is not optimal with respect to T, as assumed earlier. When s 3 (s in Proposition 1) is detered, it partitions the rectangle into four new regions. One the one hand s 3 is optimal therefore the region bounded by c Af(s 3 )B contains no other non-doated solutions; and on the other all solutions in f(s 3 )C( (s 2 ), f 2 (s 1 ))D are doated by f(s 3 ), thus the only regions that may contain other non-doated solutions are the rectangles Af(s 1 )Cf(s 3 ) and Bf(s 3 )Df(s 2 ) Figure 3.(b) which are the rectangles associated with the two 6

7 pairs of solutions (s 1, s 2 ) and (s 2, s 3 ), respectively. This procedure can be applied to each of those regions, and be repeated while there still are adjacent pairs of solutions left to analyse. In the end all the regions that might contain unsupported solutions will have been sweeped. A summary of the phase 2 method just proposed is included in Algorithm 2. Algorithm 2 (Unsupported non-doated solutions deteration within the duality gap defined by (S 1, S 2 )). // (S 1, S 2 ) is a given pair of adjacent supported solutions // L is a set that stores the calculated non-doated solutions // X is an auxiliary set that stores the calculated pairs of solutions s 1 S 1 s 2 S 2 L {s 1, s 2 } X {(s 1, s 2 )} While X Do (s 1, s 2 ) element in X; X X {(s 1, s 2 )} w 1 f 2 (s 1 ) c 2; w 2 (s 2 ) c 1 s best solution to (4) If s is defined Then L L {s} If (s) (s 2 ) Then X X {(s 1, s)} If f 2 (s) f 2 (s 1 ) Then X X {(s, s 2 )} EndIf EndWhile Finally it should be remarked that problem (4) can be formulated as the equivalent single criteria mixed integer linear program v s. t. w 1 (s) v w 1 c 1 w 2 f 2 (s) v w 2 c 2 s S (5) and thus it can be solved by means of a mixed integer programg solver. As a final remark we mention that in practice it can be difficult to compute lexicographic best solutions at the beginning of the algorithm. As a consequence the initial search area can be wider than what is actually necessary and when inserting new solutions it should be verified whether they doate some of the previous. The proposed method is suitable for an interactive approach with the decision maker, as it allows him (her) to choose which pairs of solutions to analyze, and thus which regions to further explore seeking for non-doated solutions. As an example of the method s application we consider the following bicriteria knapsack problem 7

8 with ten items, 30x x 2 89x x 4 + 6x x x x 8 23x 9 54x 10 86x 1 53x 2 33x 3 4x x 5 43x x 7 72x 8 84x 9 37x 10 (6) s. t. 12x x x x 4 + 3x x x x x x x {0, 1} 10 The images of its supported (in red) and unsupported (in green) non-doated solutions are depicted in Figure 4. f supported solution unsupported solution Figure 4: Efficient solutions with Algorithms 1 and 2 applied to problem (6) 3 Computational experiments The two phases of the algorithm were implemented in C language and used CPLEX 12.2 to solve the intermediate mixed integer programs. The tests ran on a Dual Core AMD Opteron at 2 GHz, with 4 Gb of RAM, for two sets of randomly generated instances: The first set of instances consists of integer programs with two objective functions with m = 30, 40, 50, 100 constraints and n = 2m, 4m variables. 20% of the cost coefficients c ij are generated in [ 100, 1], whereas the remaining are obtained uniformly in [0, 100]. The matrix coefficients a ij are integers distributed as 10% in [ 100, 1], 10% equal to 0, and 80% in [1, 100]. The right-hand side coefficients b i are integers calculated in [100, n j=1 a ij]. The second set of instances consists of knapsack problems again with two objective functions, of type such that (x) = c T 1 x f 2 (x) = c T 2 x ax W x {0, 1} n with c ij and a j uniformly generated in [ 100, 100], and W a random integer in [100, n j=1 a j]. 8

9 For each problem dimension thirty instances were generated, and the previous algorithms were applied. The average results in terms of the running times and of the number of computed nondoated solutions are summarized in Table 1 for the general integer problems, and in Table 2 for the knapsack problems. Table 1: Results for random integer problems CPU times (in seconds) m n 2m 2m 2m 2m 4m 4m 4m 4m Sorting Phase Phase Total Number of solutions m n 2m 2m 2m 2m 4m 4m 4m 4m Phase Phase Total The sorting time on Tables 1 and 2 refers to the time used to sort all the supported solutions found along phase 1, in order to pair adjacent solutions. For the considered instances the CPU time of this task was negligible. Similarly the CPU times of phases 1 and 2 are small, under 1 second, on general integer instances, although the number of non-doated solutions is also rather small in these cases, specially unsupported solutions. Table 2: Results for random knapsack problems CPU times (in seconds) n Sorting Phase Phase Total Number of solutions n Phase Phase Total Knapsack instances show a number of non-doated solutions greater than the general integer instances. Consequently the CPU times are greater as well. Still, for the biggest problems, comprising 100 items, 320 solutions were computed in an average time of seconds, used to find supported solutions and seconds to find unsupported solutions. 9

10 It should be noticed that the presented times refer to the problem of finding all the nondoated solutions. When applied as an interactive approach the running times of this methods would be smaller, depending on the regions to be searched. Acknowledgments This work has been partially supported by Fundação para a Ciência e a Tecnologia (FCT) under projects PEst-C/EEI/UI0308/2011 and PTDC/EEA-TEL/101884/2008. References [1] J. Clímaco and M. Pascoal. Multicriteria path and tree problems: discussion on exact algorithms and applications International Transactions in Operational Research, 19:63-98, [2] J. Cohon. Multiobjective programg and planning. Academic Press, New York, [3] J. Coutinho-Rodrigues, J. Clímaco, and J. Current. An interactive bi-objective shortest path approach: searching for unsupported non doated solutions. Computers & Operations Research, 26: , [4] J. Current, C. ReVelle, and J. Cohon. An interactive approach to identify the best compromise solution for two objective shortest path problems. Computers & Operations Research, 17: , [5] A. Raith, and M. Ehrgott. A comparison of solution strategies for biobjective shortest path problems. Computers & Operations Research, 36: , [6] S. Steiner, and T. Radzik. Computing all efficient solutions of the biobjective imum spanning tree problem. Computers & Operations Research, 35: , [7] R. Steuer. Multiple Criteria Optimization. Theory, Computation and Application. John Wiley, New York, [8] R. Steuer and E.-U. Choo, An interactive weighted Tchebycheff procedure for multiple objective programg. Mathematical Programg, 26: ,

### Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC Coimbra

Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC Coimbra João Clímaco, M. Eugénia Captivo and Marta Pascoal On the bicriterion minimum

### Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers

Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers IESC - Coimbra Carlos Gomes da Silva and Pedro M.R. Carreira Selecting Audit Targets Using Benford

### Core problems in the bi-criteria {0,1}-knapsack: new developments

Core problems in the bi-criteria {0,}-knapsack: new developments Carlos Gomes da Silva (2,3, ), João Clímaco (,3) and José Figueira (3,4) () Faculdade de Economia da Universidade de Coimbra Av. Dias da

### A hierarchical multicriteria routing model with traffic splitting for MPLS networks

A hierarchical multicriteria routing model with traffic splitting for MPLS networks João Clímaco, José Craveirinha, Marta Pascoal jclimaco@inesccpt, jcrav@deecucpt, marta@matucpt University of Coimbra

### Instituto de Engenharia de Sistemas e Computadores de Coimbra. Institute of Systems Engineering and Computers INESC - Coimbra

Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC - Coimbra Carlos Simões, Teresa Gomes, José Craveirinha, and João Clímaco A Bi-Objective

### Traffic splitting in MPLS networks a hierarchical multicriteria approach

Paper Traffic splitting in MPLS networks a hierarchical multicriteria approach José M. F. Craveirinha, João C. N. Clímaco, Marta M. B. Pascoal, and Lúcia M. R. A. Martins Abstract In this paper we address

### A Reference Point Method to Triple-Objective Assignment of Supporting Services in a Healthcare Institution. Bartosz Sawik

Decision Making in Manufacturing and Services Vol. 4 2010 No. 1 2 pp. 37 46 A Reference Point Method to Triple-Objective Assignment of Supporting Services in a Healthcare Institution Bartosz Sawik Abstract.

### Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows

TECHNISCHE UNIVERSITEIT EINDHOVEN Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows Lloyd A. Fasting May 2014 Supervisors: dr. M. Firat dr.ir. M.A.A. Boon J. van Twist MSc. Contents

### Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC - Coimbra

Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC - Coimbra José L. Sousa, Humberto M. Jorge, António G. Martins A multi-objective evolutionary

### 5. Convergence of sequences of random variables

5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

### NEW VERSION OF DECISION SUPPORT SYSTEM FOR EVALUATING TAKEOVER BIDS IN PRIVATIZATION OF THE PUBLIC ENTERPRISES AND SERVICES

NEW VERSION OF DECISION SUPPORT SYSTEM FOR EVALUATING TAKEOVER BIDS IN PRIVATIZATION OF THE PUBLIC ENTERPRISES AND SERVICES Silvija Vlah Kristina Soric Visnja Vojvodic Rosenzweig Department of Mathematics

### A Weighted-Sum Mixed Integer Program for Bi-Objective Dynamic Portfolio Optimization

AUTOMATYKA 2009 Tom 3 Zeszyt 2 Bartosz Sawik* A Weighted-Sum Mixed Integer Program for Bi-Objective Dynamic Portfolio Optimization. Introduction The optimal security selection is a classical portfolio

### Load Balancing of Telecommunication Networks based on Multiple Spanning Trees

Load Balancing of Telecommunication Networks based on Multiple Spanning Trees Dorabella Santos Amaro de Sousa Filipe Alvelos Instituto de Telecomunicações 3810-193 Aveiro, Portugal dorabella@av.it.pt Instituto

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### INTERACTIVE SEARCH FOR COMPROMISE SOLUTIONS IN MULTICRITERIA GRAPH PROBLEMS. Lucie Galand. LIP6, University of Paris 6, France

INTERACTIVE SEARCH FOR COMPROMISE SOLUTIONS IN MULTICRITERIA GRAPH PROBLEMS Lucie Galand LIP6, University of Paris 6, France Abstract: In this paper, the purpose is to adapt classical interactive methods

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, In which we introduce the theory of duality in linear programming.

Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, 2011 Lecture 6 In which we introduce the theory of duality in linear programming 1 The Dual of Linear Program Suppose that we

### The multi-integer set cover and the facility terminal cover problem

The multi-integer set cover and the facility teral cover problem Dorit S. Hochbaum Asaf Levin December 5, 2007 Abstract The facility teral cover problem is a generalization of the vertex cover problem.

### Planar Tree Transformation: Results and Counterexample

Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem

### Convex Programming Tools for Disjunctive Programs

Convex Programming Tools for Disjunctive Programs João Soares, Departamento de Matemática, Universidade de Coimbra, Portugal Abstract A Disjunctive Program (DP) is a mathematical program whose feasible

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Good luck, veel succes!

Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

### Optimal Scheduling for Dependent Details Processing Using MS Excel Solver

BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 8, No 2 Sofia 2008 Optimal Scheduling for Dependent Details Processing Using MS Excel Solver Daniela Borissova Institute of

### COMMUNICATING LIFE-CYCLE ASSESSMENT RESULTS: A COMPARISON OF VEHICLE ALTERNATIVES IN PORTUGAL

Energy for Sustainability 2013 Sustainable Cities: Designing for People and the Planet Coimbra, 8 to 10 September, 2013 COMMUNICATING LIFE-CYCLE ASSESSMENT RESULTS: A COMPARISON OF VEHICLE ALTERNATIVES

### Rita Girão-Silva 1,2*, José Craveirinha 1,2 and João Clímaco 2

Girão-Silva et al. Journal of Uncertainty Analysis and Applications 2014, 2:3 RESEARCH Open Access Stochastic hierarchical multiobjective routing model in MPLS networks with two service classes: an experimental

### Chapter 2 Goal Programming Variants

Chapter 2 Goal Programming Variants This chapter introduces the major goal programming variants. The purpose and underlying philosophy of each variant are given. The three major variants in terms of underlying

### Fast Sequential Summation Algorithms Using Augmented Data Structures

Fast Sequential Summation Algorithms Using Augmented Data Structures Vadim Stadnik vadim.stadnik@gmail.com Abstract This paper provides an introduction to the design of augmented data structures that offer

UNIVERSIDADE DE SÃO PAULO Instituto de Ciências Matemáticas e de Computação ISSN 0103-2569 Comments on On minimizing the lengths of checking sequences Adenilso da Silva Simão N ō 307 RELATÓRIOS TÉCNICOS

### 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

### Min-cost flow problems and network simplex algorithm

Min-cost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.

### 1 Polyhedra and Linear Programming

CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

### GRAPH THEORY and APPLICATIONS. Trees

GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.

### 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

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### Lecture 1: Linear Programming Models. Readings: Chapter 1; Chapter 2, Sections 1&2

Lecture 1: Linear Programming Models Readings: Chapter 1; Chapter 2, Sections 1&2 1 Optimization Problems Managers, planners, scientists, etc., are repeatedly faced with complex and dynamic systems which

### 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

### A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem

A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,

### Using the Simplex Method in Mixed Integer Linear Programming

Integer Using the Simplex Method in Mixed Integer UTFSM Nancy, 17 december 2015 Using the Simplex Method in Mixed Integer Outline Mathematical Programming Integer 1 Mathematical Programming Optimisation

### Trees and Fundamental Circuits

Trees and Fundamental Circuits Tree A connected graph without any circuits. o must have at least one vertex. o definition implies that it must be a simple graph. o only finite trees are being considered

### The Bi-Objective Pareto Constraint

The Bi-Objective Pareto Constraint Renaud Hartert and Pierre Schaus UCLouvain, ICTEAM, Place Sainte Barbe 2, 1348 Louvain-la-Neuve, Belgium {renaud.hartert,pierre.schaus}@uclouvain.be Abstract. Multi-Objective

### MODULE 15 Clustering Large Datasets LESSON 34

MODULE 15 Clustering Large Datasets LESSON 34 Incremental Clustering Keywords: Single Database Scan, Leader, BIRCH, Tree 1 Clustering Large Datasets Pattern matrix It is convenient to view the input data

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### Massive Data Classification via Unconstrained Support Vector Machines

Massive Data Classification via Unconstrained Support Vector Machines Olvi L. Mangasarian and Michael E. Thompson Computer Sciences Department University of Wisconsin 1210 West Dayton Street Madison, WI

### Scheduling the Brazilian soccer tournament with fairness and broadcast objectives

Scheduling the Brazilian soccer tournament with fairness and broadcast objectives Celso C. Ribeiro 1 and Sebastián Urrutia 2 1 Department of Computer Science, Universidade Federal Fluminense, Rua Passo

### Social Media Mining. Graph Essentials

Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

### A Study of Local Optima in the Biobjective Travelling Salesman Problem

A Study of Local Optima in the Biobjective Travelling Salesman Problem Luis Paquete, Marco Chiarandini and Thomas Stützle FG Intellektik, Technische Universität Darmstadt, Alexanderstr. 10, Darmstadt,

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### Offline sorting buffers on Line

Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

### KEYWORD SEARCH OVER PROBABILISTIC RDF GRAPHS

ABSTRACT KEYWORD SEARCH OVER PROBABILISTIC RDF GRAPHS In many real applications, RDF (Resource Description Framework) has been widely used as a W3C standard to describe data in the Semantic Web. In practice,

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### Project selection with limited resources in data envelopment analysis

Available online at http://ijim.srbiau.ac.ir/ Int. J. Industrial Mathematics (ISSN 2008-5621) Vol. 7, No. 1, 2015 Article ID IJIM-00564, 6 pages Research Article Project selection with limited resources

### a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

### Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the

### 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

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### ON THE TREE GRAPH OF A CONNECTED GRAPH

Discussiones Mathematicae Graph Theory 28 (2008 ) 501 510 ON THE TREE GRAPH OF A CONNECTED GRAPH Ana Paulina Figueroa Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria,

### Minimum Caterpillar Trees and Ring-Stars: a branch-and-cut algorithm

Minimum Caterpillar Trees and Ring-Stars: a branch-and-cut algorithm Luidi G. Simonetti Yuri A. M. Frota Cid C. de Souza Institute of Computing University of Campinas Brazil Aussois, January 2010 Cid de

### Supply Chain Design and Inventory Management Optimization in the Motors Industry

A publication of 1171 CHEMICAL ENGINEERING TRANSACTIONS VOL. 32, 2013 Chief Editors: Sauro Pierucci, Jiří J. Klemeš Copyright 2013, AIDIC Servizi S.r.l., ISBN 978-88-95608-23-5; ISSN 1974-9791 The Italian

### Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

### The Method of Lagrange Multipliers

The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### Instituto de Engenharia de Sistemas e Computadores Institute of Systems Engineering and Computers INESC Coimbra

ISSN 1645-4847 Instituto de Engenharia de Sistemas e Computadores Institute of Systems Engineering and Computers INESC Coimbra IRIS - Interactive Robustness analysis and parameters' Inference for multicriteria

### Monotone Partitioning. Polygon Partitioning. Monotone polygons. Monotone polygons. Monotone Partitioning. ! Define monotonicity

Monotone Partitioning! Define monotonicity Polygon Partitioning Monotone Partitioning! Triangulate monotone polygons in linear time! Partition a polygon into monotone pieces Monotone polygons! Definition

### A MILP Scheduling Model for Multi-stage Batch Plants

A MILP Scheduling Model for Multi-stage Batch Plants Georgios M. Kopanos, Luis Puigjaner Universitat Politècnica de Catalunya - ETSEIB, Diagonal, 647, E-08028, Barcelona, Spain, E-mail: luis.puigjaner@upc.edu

### 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

### Solving Method for Linear Fractional Optimization Problem with Fuzzy Coefficients in the Objective Function

INT J COMPUT COMMUN, ISSN 1841-9836 8(1):146-152, February, 2013. Solving Method for Linear Fractional Optimization Problem with Fuzzy Coefficients in the Objective Function B. Stanojević, M. Stanojević

### Flow and Activity Analysis

Facility Location, Layout, and Flow and Activity Analysis Primary activity relationships Organizational relationships» Span of control and reporting hierarchy Flow relationships» Flow of materials, people,

### 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

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### Absolute Value Programming

Computational Optimization and Aplications,, 1 11 (2006) c 2006 Springer Verlag, Boston. Manufactured in The Netherlands. Absolute Value Programming O. L. MANGASARIAN olvi@cs.wisc.edu Computer Sciences

### 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

### R u t c o r Research R e p o r t. A Method to Schedule Both Transportation and Production at the Same Time in a Special FMS.

R u t c o r Research R e p o r t A Method to Schedule Both Transportation and Production at the Same Time in a Special FMS Navid Hashemian a Béla Vizvári b RRR 3-2011, February 21, 2011 RUTCOR Rutgers

### A Maximal Covering Model for Helicopter Emergency Medical Systems

The Ninth International Symposium on Operations Research and Its Applications (ISORA 10) Chengdu-Jiuzhaigou, China, August 19 23, 2010 Copyright 2010 ORSC & APORC, pp. 324 331 A Maximal Covering Model

### 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

### CS 2336 Discrete Mathematics

CS 2336 Discrete Mathematics Lecture 5 Proofs: Mathematical Induction 1 Outline What is a Mathematical Induction? Strong Induction Common Mistakes 2 Introduction What is the formula of the sum of the first

### 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

### Arrangements And Duality

Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,

Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

### Multiple Resource Constraint Time-Cost-Resource Optimization Using Genetic Algorithm

First International Conference on Construction In Developing Countries (ICCIDC I) Advancing and Integrating Construction Education, Research & Practice August -,, Karachi,, Pakistan Multiple Resource Constraint

### Random Geometric Graph Diameter in the Unit Disk with l p Metric (Extended Abstract)

Random Geometric Graph Diameter in the Unit Disk with l p Metric (Extended Abstract) Robert B. Ellis 1,, Jeremy L. Martin 2,, and Catherine Yan 1, 1 Department of Mathematics, Texas A&M University, College

### ABOUT UNIQUELY COLORABLE MIXED HYPERTREES

Discussiones Mathematicae Graph Theory 20 (2000 ) 81 91 ABOUT UNIQUELY COLORABLE MIXED HYPERTREES Angela Niculitsa Department of Mathematical Cybernetics Moldova State University Mateevici 60, Chişinău,

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

### 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

### 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

### EXCEL SOLVER TUTORIAL

ENGR62/MS&E111 Autumn 2003 2004 Prof. Ben Van Roy October 1, 2003 EXCEL SOLVER TUTORIAL This tutorial will introduce you to some essential features of Excel and its plug-in, Solver, that we will be using

### Locating and sizing bank-branches by opening, closing or maintaining facilities

Locating and sizing bank-branches by opening, closing or maintaining facilities Marta S. Rodrigues Monteiro 1,2 and Dalila B. M. M. Fontes 2 1 DMCT - Universidade do Minho Campus de Azurém, 4800 Guimarães,

### AXIOMS FOR INVARIANT FACTORS*

PORTUGALIAE MATHEMATICA Vol 54 Fasc 3 1997 AXIOMS FOR INVARIANT FACTORS* João Filipe Queiró Abstract: We show that the invariant factors of matrices over certain types of rings are characterized by a short

### 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

### Big Data Techniques Applied to Very Short-term Wind Power Forecasting

Big Data Techniques Applied to Very Short-term Wind Power Forecasting Ricardo Bessa Senior Researcher (ricardo.j.bessa@inesctec.pt) Center for Power and Energy Systems, INESC TEC, Portugal Joint work with

### Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams

Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams André Ciré University of Toronto John Hooker Carnegie Mellon University INFORMS 2014 Home Health Care Home health care delivery

### LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA Hyderabad

LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA 98481 85073 Hyderabad Page 1 of 19 Question: Explain LPP. Answer: Linear programming is a mathematical technique for determining the optimal allocation of resources

### 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

### The distance of a curve to its control polygon J. M. Carnicer, M. S. Floater and J. M. Peña

The distance of a curve to its control polygon J. M. Carnicer, M. S. Floater and J. M. Peña Abstract. Recently, Nairn, Peters, and Lutterkort bounded the distance between a Bézier curve and its control

### A Constraint Programming based Column Generation Approach to Nurse Rostering Problems

Abstract A Constraint Programming based Column Generation Approach to Nurse Rostering Problems Fang He and Rong Qu The Automated Scheduling, Optimisation and Planning (ASAP) Group School of Computer Science,