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ISE 336 ART OF MATHEMATICAL MODELLING INTRODUCTION TO MATHEMATICAL MODELLING AND IBM ILOG CPLEX OPTIMIZATION STUDIO 2014-2015 SPRING TERM Dr. M. Arslan Örnek LECTURE#1 OBJECTIVES • Developing mathematical models and heuristic methods for well known industrial problems • Reading and understanding existing models, algorithms • Mathematical Modeling and programming using IBM ILOG CPLEX OPTIMIZATION STUDIO 2 WHY DO WE NEED MODELS ? Complicated practical problems can be investigated by using models, which reflect the relevant features of the problem, in such a way that properties of the model can be related to the real (actual) situation. 3 MODELING CONCEPT 4 WHY IS MODEL BUILDING AN ART ? Designing and implementing of mathematical models demands not only the appropriate technical and mathematical knowledge of the modeller, but also requires high social skills as well. 5 WHY IS MODEL BUILDING AN ART ? • Knowledge of a modeling language (OPL, GAMS, LINGO etc. ) does not make someone a modeler any more than knowing MATLAB makes one a mathematician. • Mathematical modeling is both an art and a science. • Coding the model in a modeling language according to the mathematical structure (linear, integer, nonlinear) of the problem is the science part. • But the analysis and defining/constructing modeling components of a system are the art. • For example, questions such as how much detail to include in the model or how to represent a certain phenomena (i.e., interaction between system components) are all a part of the art. 6 BASIC TERMS • Mathematical Model is the collection of variables and relationships to describe a problem. • Operations Research is the study of how to form mathematical models of complex engineering and management problems and how to analyze them to gain insight about possible solutions. • Mathematical Program (Optimization Models) can be defined as a mathematical representation aimed at programming or planning the best possible allocation of scarce resources. Representation includes : • choices as decision variables and • determine values that maximize or minimize objective function of the decision variables • subject to constraints on variable values expressing the limits on possible decision choices. 7 MATHEMATICAL PROGRAMMING STEPS 8 TOOLS FOR MODEL TRANSLATION GENERAL PURPOSE LANGUAGES (C/C++, VISUAL BASIC, JAVA) –Tedious, low-level, error-prone –But, almost complete flexibility MATHEMATICAL PROGRAMMING LANGUAGES –LINGO,GAMS,IBM ILOG CPLEX Optimization Studio –Easy to use, you don’t have to deal with low-level coding 9 IBM ILOG CPLEX OPTIMIZATION STUDIO 10 11 APPLICATION AREAS • Linear Programming : –Product mix –Make-buy –Media selection –Marketing research –Portfolio selection –Shipping & transportation –... 12 APPLICATION AREAS • Integer Programming : –Capital Budgeting –Warehouse Location –Scheduling –Either-or decisions –... 13 APPLICATION AREAS • Nonlinear Programming : –Capital Budgeting –Data networks and analysis –Resource allocation –Computer-aided design –Least squares formulations –Modeling human or organization behavior –Process and power industries –Quantitative finance 14 SUCCESSFUL INTEGER PROGRAMMING APPLICATIONS See the following links for more details and examples http://interfaces.journal.informs.org/content/by/year http://www.sciencedirect.com/ http://springerlink.com/ http://www.informaworld.com/ http://search.ebscohost.com http://scholar.google.com.tr/ 15 BUILDING A MATHAMATICAL MODEL : DECISION VARIABLES, CONSTRAINTS, OBJECTIVE FUNCTIONS The standard statement of an optimization model has the form : Min or max (objective function) Subject to (constraints) 16 GENERAL MATHEMATICAL PROGRAMMING FORMAT 17 A linear programming problem (LP) is a class of the mathematical programming problem, a constrained optimization problem, in which we seek to find a set of values for continuous variables (x1, x2, . . . ,xn) that maximizes or minimizes a linear objective function z, while satisfying a set of linear constraints (a system of simultaneous linear equations and/or inequalities). Mathematically, an LP is expressed as follows: 18 An integer (linear) programming problem (IP) is a linear programming problem in which at least one of the variables is restricted to integer values. 19 20 • The term "programming" in this context means planning activities that consume resources and/or meet requirements, as expressed in the m constraints, not the other meaning— coding computer programs. • The resources may include raw materials, machines, equipments, facilities, workforce, money, management, information technology, and so on. In the real world, these resources are usually limited and must be shared with several competing activities. Requirements may be implicitly or explicitly imposed. • The objective of the LP/IP is to allocate the shared resources, and responsibility to meet requirements, to all competing activities in an optimal (best possible) manner. 21 A mixed integer program will be said to be in standard form if (1) the objective function is maximized, (2) all the constraints are of ≤ form, (3) each integer variable is defined over consecutive integer numbers whose lower bound is 0 and upper bound infinity, and (4) each continuous variable is nonnegative with no finite upper bound. Any MIP that does not conform to the conditions (l)-(4) is considered to be in nonstandard form, but may be converted to a standard one through simple mathematical manipulations. The following are various nonstandard forms that need to be converted: • Minimization problem • Inequality of ≥ form • Equation (equality constraint) • Unrestricted variable (continuous or integer) • Variable with a positive or a negative lower bound • Variable with a finite upper bound 22 23 • A combinatorial optimization problem (COP) is a discrete optimization problem in which we seek to find a solution in a finite set of solutions that maximizes or minimizes an objective function. This type of problem usually arises in the selection of a finite set of mutually exclusive alternatives. These qualitative alternatives may be quantified by the use of discrete variables. Usually, the set of all possible solutions can be enumerated and their associated objective values can be evaluated to determine an optimum solution. But unfortunately, the number of solutions by complete enumeration is usually too huge even for a moderate-sized problem. • Well-known examples of COP include the classical assignment problem and traveling salesman problem (TSP). The assignment problem may be applied, for example, to assign n jobs to n workers in a most efficient manner so that each job is assigned to one and only one worker, and vice versa. The TSP originates from a salesman who starts from a home city to visit n — 1 cities so that each city is visited once and only once and then returns to the home city with a minimum travel distance. 24 BUILDING A BASIC MODEL 25 SYSTEM BOUNDARIES 26 INPUT PARAMETERS Input Parameters : Quantities that we will take as fixed during the decision process • Cost • Availabilities • Yields • Requirements 27 DECISION VARIABLES Decision variables : • Variables in optimization models represent the decisions to be taken. X1: barrels of Saudi crude refined per day (in thousands) X2: barrels of Venezuelan crude refined per day (in thousands) 28 OBJECTIVE FUNCTIONS • Objective function in an optimization model quantify the decision consequences to be maximized or minimized. 29 CONSTRAINTS • Variable type constraints specify the domain of definition for decision variables: the set of values which the variables have meaning 30 CONSTRAINTS • Main constraints of optimization models specify the restrictions and interactions, other than variable type, that limit decision variable values. 31 CRUDE OIL CONCEPTUAL MODEL Minimize Total purchasing cost (1) Subject to Meet customer demand for each type of product (2) Ensure supply capacity for each supply point (3) Nonnegativity constraints for each decision variables (4) 32 CRUDE OIL MATHEMATICAL MODEL 33 LARGE SCALE OPTIMIZATION AND INDEXING • In contrast to Two Crude Petroleum example, in real applications, optimization models quickly grow to thousands, even millions, of variables and constraints. • Therefore, now we’ll see the indexed notational representation that keep large models manageable. 34 INDEXING : THE FIRST STEP • The first step in formulating a large optimization model is to choose appropriate indexes for the different dimensions of the problem. • Indexes or subscripts permit representing collections of similar quantities with a single symbol. For example, represents 100 similar values with the same name z, distinguishing them with the index i. 35 INDEXED DECISION VARIABLES • Indexing makes it possible to define a large number of variables with a few symbols. Suppose that an optimization model employs decision variables wijkl where i and k range over 1,...,100, while j and l index through 1,...,50. Therefore the total number of variables which we represent with wijkl is 25,000,000 (100 x 50 x 100 x 50 ) 36 INDEXED SYMBOLIC PARAMETERS • To describe large-scale optimization models compactly, it is usually necessary to assign indexed symbolic names to most input parameters, even though they are being treated as constant. • Furthermore, summation notation may be employed for representation. 37 INDEXED SYMBOLIC PARAMETERS-EXAMPLES (a) Write the following sum more compactly with summation notation : (b) Write out terms of the sum separately : 38 INDEXED SYMBOLIC PARAMETERS-EXAMPLES 39 INDEXED FAMILIES OF CONSTRAINTS • We could write each constraint explicitly, but when the large size models are considered, the model would become very bulky and hard to comprehend. • Mathematical programming notation deals with this difficulty by listing indexed families of constraints. 40 INDEXED FAMILIES OF CONSTRAINTS • Families of similar constraints distinguished by indexes may be expressed in a single line format. (Constraint for fixed indexes) (range of indexes) which implies one constraint for each combination of indexes in the ranges specified. 41 INDEXED FAMILIES OF CONSTRAINTS-EXAMPLE An optimization model must decide how to allocate available supplies si at sources i = 1,...,m to meet requirements rj at customers j = 1,...,n using variables wij: amount allocated from source i to customer j Formulate each of the following requirements in a single line: 42 INDEXED FAMILIES OF CONSTRAINTS-EXAMPLE (a)The amount allocated from source 32 cannot exceed the supply available at 32. (b)The amount allocated from each source i cannot exceed the supply available at i. (c)The amount allocated to customer n should equal the requirement at n. (d)The amount allocated to each customer j should equal the requirement at j. 43 INDEXED FAMILIES OF CONSTRAINTS-EXAMPLE 44 INDEXED FAMILIES OF CONSTRAINTS-EXAMPLE 45 VOLSAY GAS PRODUCTION PLAN Consider a Belgian company Volsay, which specializes in producing ammoniac gas (N H3) and ammonium chloride (N H4 Cl). Volsay has at its disposal 50 units of nitrogen (N), 180 units of hydrogen (H), and 40 units of chlorine (Cl). The company makes a profit of 40 Belgian francs for each sale of an ammoniac gas unit and 50 Belgian francs for each sale of an ammonium chloride unit. Volsay would like a production plan to maximize its profits given its available stocks. 46 VOLSAY GAS PRODUCTION PLAN 47 VOLSAY MATHEMATICAL MODEL Objective : Find a production plan that maximizes profit Constraints : Available inventory quantity for components 48 VOLSAY MATHEMATICAL MODEL DEFINITIONS 49 VOLSAY MATHEMATICAL MODEL OPEN FORM 50 VOLSAY MATHEMATICAL MODEL COMPACT FORM 51