Dr. Arslan Ornek MATHEMATICAL MODELS 9/17/2019 CHAPTER ONE MATHEMATICAL MODELS 9/17/2019 ISE420 Algorithmic Operations Research ISE420 Algorithmic Operations Research

We have “operations” problems: planning work shifts choosing investments for available funds designing facilities for customer service determining product mix, etc. can be formulated as a mathematical model 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research A MATHEMATICAL MODEL is the collection of variables and relationships needed to describe pertinent features of such a problem. 1.1 Operations Research (OR) 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. 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research OR deals with “decision problems” by formulating and analyzing mathematical models – mathematical representations of pertinent problem features. Inference Assessment Analysis Modeling Problem Model Decisions Conclusions 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research Modeling: Define variables and quantify the relationships needed to describe relevant system behavior. Analysis: Apply mathematical skills and technology to see what conclusions the model suggests. Inference: Argue that conclusions drawn from the model are meaningful enough to infer decisions. Assessment: Evaluate decisions for further thought that might lead to revised modeling. 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research 1.2 The three fundamental concerns in forming OR models are the DECISIONS open to decision makers the CONSTRAINTS limiting decision choices, and the OBJECTIVES making some decisions preferred to others. 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research 1.3 OPTIMIZATION MODELS (also called MATHEMATICAL PROGRAMS) represent problem choices as decision variables and seek values that maximize or minimize objective functions of the decision variables subject to constraints on variable values expressing the limits on possible decision choices. 1.4 A FEASIBLE SOLUTION is a choice of values for the decision variables that satisfies all constraints. OPTIMAL SOLUTIONS are feasible solutions that achieve objective function value(s) as good as those of any other feasible solutions. 9/17/2019 ISE420 Algorithmic Operations Research

A line between those items taken as settled and those to be decided is called the SYSTEM BOUNDARY. Parameters – quantities taken as given –define objective functions and constraints applicable to the decision model inside. Together, parameters and decision variables determine results measured as output variables. System boundary Output variables Parameters Modeling and analysis in the decision variables 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research Ex. Simple inventory model to determine when and how much to order Parameters: d: demand, f: fixed cost of replenishment, h: holding cost, s: cost of lost sales l: lead time, m: min. order size Decision variables : q: reorder quantity r: reorder point Output variable: c: total cost of ordering, holding and lost sales Objective function Min c s.t. constraints r => l d q=> m or q=> l d 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research 1.6 SENSITIVITY ANALYSIS is an exploration of results from mathematical models to evaluate how they depend on the values chosen for parameters. Solutions prescribing choices for decision variables by simple formulas in the input variables are called closed-form solutions. 1.7 CLOSED-FORM solutions represent the ultimate in analysis of mathematical models because they provide both immediate results and rich sensitivity analysis. 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research 1.8 TRACTABILITY in modeling means the degree to which the model admits convenient analysis –how much analysis is practical. 1.9 The VALIDITY of a model is the degree to which inferences drawn from the model hold for the real system. 1.10 OR analysts almost always confront a trade-off between validity of models and their tractability to analysis. 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research A SIMULATION MODEL is a computer program that simply steps through the behaviour of a system of interest and reports experience. 1.11 Simulation models often posses high validity because they track true system behavior fairly accurately. Models that evaluate fixed decision alternatives rather indicating good choices may be termed DESCRIPTIVE MODELS. 1.12 Descriptive models yield fewer analytic inferences than prescriptive, optimization models because they take both input parameters and decisions as fixed. 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research Numerical Search is the process of systematically trying different choices for the decision variables, keeping track of the feasible one with the best objective function value found so far. 1.13 Inferences from numerical search are limited to the specific points explored unless mathematical structure in the model supports further deduction. 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research 1.14 An EXACT OPTIMAL SOLUTION is a feasible solution to an optimization model that is provably as good as any other in the objective function value. A HEURISTIC or APPROXIMATE OPTIMUM is a feasible solution derived from prescriptive analysis that is not guaranteed to yield an exact optimum. 1.15 Losses from settling for heuristic instead of exact optimal solutions are often dwarfed by variations associated with questionable model assumptions and doubtful data. 1.16 The appeal of exact optimal solutions is that they provide both good feasible solutions and certainty about what can be achieved under a fixed set of model assumptions. 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research 1.17 A mathematical model is termed DETERMINISTIC if all parameter values are assumed to be known with certainty, and PROBABILISTIC or STOCHASTIC if it involves quantities known only in probability. RANDOM VARIABLES represent quantities known only in terms of a probability in stochastic models. 1.18 Besides providing only descriptive analysis, stochastic simulation models impose the extra analytic burden of having to estimate results statistically from a simple of system realizations. 9/17/2019 ISE420 Algorithmic Operations Research

ISE420 Algorithmic Operations Research 1.19 The power and generality of available mathematical tools for analysis of stochastic models does not nearly match that available for deterministic models. 1.20 Most optimization models are deterministic - not because OR analysts really believe that all problem parameters are known with certainty, but because useful prescriptive results can often be obtained only if stochastic variation is ignored. 1.21 The model – based OR approach to problem solving works best on problems important enough to warrant the time and resources for a careful study. 9/17/2019 ISE420 Algorithmic Operations Research