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CHAPTER TWO

CHAPTER TWO. DETERMINISTIC OPTIMIZATION MODELS.

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CHAPTER TWO

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  1. CHAPTER TWO DETERMINISTIC OPTIMIZATION MODELS ISE420 Algorithmic Operations Research

  2. Deterministic Optimization Models are also called MATHEMATICAL PROGRAMS because they decide how to plan or program activities.The first step in formulating any optimization model is to identify the DECISION VARIABLES.2.1 Variables in optimization models represent the decisions to be taken. ISE420 Algorithmic Operations Research

  3. The next issue in formulating any optimization model is constraints. What limits decisions?2.2 VARIABLE TYPE CONSTRAINTS specify the domain of definition for decision variables: the set of values for which the variables have meaning.The most common variable – type constraint form: nonnegativity.2.3 MAIN CONSTRAINTS of optimization models specify the restrictions and interactions, other than variable type, that limit decision variable values. ISE420 Algorithmic Operations Research

  4. OBJECTIVE or CRITERION (evaluation) FUNCTIONS tell us how to rate decisions.2.4 Objective functions in optimization models quantify the decision consequences to be maximized or minimized.2.5 The standard statement of an optimization model has the formMin or max (objective function(s))s.t. main constraints variable - type constraints ISE420 Algorithmic Operations Research

  5. 2.6 The FEASIBLE SET (or REGION) of an optimization model is the collection of choices for decision variables satisfying all model constraints.2.12 An OPTIMAL SOLUTION is a feasible choice for decision variables with objective function value at least equal to that of any other solution satisfying all constraints.2.14 The OPTIMAL VALUE in an optimization model is the objective function value of anyoptimal solutions.2.15 An optimization model can have only one optimal value. ISE420 Algorithmic Operations Research

  6. 2.16 An optimization model may have a UNIQUE OPTIMAL SOLUTION or several ALTERNATIVE OPTIMAL SOLUTIONS.Still, all must have the same optimal value, infeasible models have no optimal solutions.2.18 An optimization model is INFEASIBLE if no choice of decision variables satisfies all constraints. ISE420 Algorithmic Operations Research

  7. Another case where an optimization model has no feasible solution arises when it is unbounded.2.20 An optimization model is UNBOUNDED when feasible choices of the decision variables can produce arbitrarily good objective function values.Unbounded models have no optimal solutions because any possibility can be improved. ISE420 Algorithmic Operations Research

  8. LINEAR AND NONLINEAR PROGRAMS2.27 The general form of a MATHEMATICAL PROGRAM or (single objective) optimization model is:Min or max f(x1,…,xn)s.t. Where f, gi are given functions of decision variables x1,…,xn and b1,…,bm are specified constant parameters. ISE420 Algorithmic Operations Research

  9. 2.28 A function is LINEAR if it is a constant - weighted sum of decision variables. Otherwise it is NONLINEAR.2.29 An optimization model in functional form 2.27 is a LINEAR PROGRAM (LP) if the (single) objective function f and all constraints functions g1,…,gm are linear in the decisionvariables. Also, decision variables should be able to take on whole number or fractional values. ISE420 Algorithmic Operations Research

  10. 2.30 An optimization model in functional form 2.27 is a NONLINEAR PROGRAM (NLP) if the (single) objective function f or any of the constraint functions g1,…,gm is nonlinear in the decision variables Also, decision variables should be able to take on whole-number or fractional values.2.32 Linear functions implicitly assume that each unit increase in a decision variable has the same effect as the preceding increase: EQUAL RETURNS TO SCALE.Nonlinear functions have the effect of DECREASING/INCREASING RETURNS TO SCALE. ISE420 Algorithmic Operations Research

  11. DISCRETE OR INTEGER PROGRAMS2.33 A variable is DISCRETE if it is limited to a fixed or countable set of values. Often, the choices are only 0 and 1.2.34 A variable is CONTINUOUS if it can take on any value in a specified interval. ISE420 Algorithmic Operations Research

  12. 2.36 An optimization model is an INTEGER PROGRAM (IP) if anyone of its decision variables is discrete.If all variables are discrete, the model is a pure integer program, otherwise it is a mixed – integer program.2.37 A discrete integer programming model is an INTEGER LINEAR PROGRAM (ILP) if its (single) objective function and all main constraints are linear.2.38 A discrete or integer programmingmodel is an INTEGER NONLINEAR PROGRAM (INLP) if its (single) objective function or any of its main constraintsis nonlinear. ISE420 Algorithmic Operations Research

  13. MULTIOBJECTIVE OPTIMIZATION MODELSA MULTIOBJECTIVE OPTIMIZATION MODEL is that maximizes or minimizes more thanone objective function at the same time. ISE420 Algorithmic Operations Research

  14. Linear Programming • minimize or maximize a linear objective • subject to linear equalities and inequalities ISE420 Algorithmic Operations Research

  15. Example: Max is in a pie eating contest that lasts 1 our. Each torte that he eats takes 2 minutes. Each apple pie that he eats takes 3 minutes. He receives 4 points for each torte and 5 points for each pie. What should Max eat so as to get the most points? ISE420 Algorithmic Operations Research

  16. Step 1. Determine the decision variables Let x be the number of tortes eaten by Max. Let y be the number of pies eaten by Max. • Step 2. Determine the objective function • Step 3. Determine the constraints Maximize z = 4x + 5y (objective function) S.t. 2x + 3y ≤60 (constraint) x ≥0 ; y ≥0 (non-negativity constraints) ISE420 Algorithmic Operations Research

  17. A feasible solution satisfies all of the constraints. • x = 10, y = 10 is feasible; • x = 10, y = 15 is infeasible. • An optimal solution is the best feasible solution. • The optimal solution is • x = 30, y = 0. ISE420 Algorithmic Operations Research

  18. Terminology • Decision variables: e.g., x and y. – In general, these are quantities you can control to improve your objective which should completely describe the set of decisions to be made. • Constraints:e.g., 2x + 3y≤24, x≥0,y≥0 – Limitations on the values of the decision variables. • Objective Function:e.g., 4x + 5y – Value measure used to rank alternatives – Seek to maximize or minimize this objective – examples: maximize NPV, minimize cost ISE420 Algorithmic Operations Research

  19. Addressing managerial problems: A management science (OR) framework 1. Determine the problem to be solved 2. Observe the system and gather data 3. Formulate a mathematical model of the problem and any important subproblems 4. Verify the model and use the model for prediction or analysis 5. Select a suitable alternative 6. Present the results to the organization 7. Implement and evaluate ISE420 Algorithmic Operations Research

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