L3_10 - ESI 6314 Deterministic Methods in Operations...

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1/1 ESI 6314 Deterministic Methods in Operations Research Lecture Notes 3 (cont.) University of Florida Department of Industrial and Systems Engineering / REEF
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2/1 A simple example Giapetto’s, Inc., manufactures wooden soldiers and trains. Each soldier built: Sells for $27 and uses $10 worth of raw materials. Increases Giapetto’s variable labor/overhead costs by $14. Requires 2 hours of finishing labor. Requires 1 hour of carpentry labor. Each train built: Sells for $21 and uses $9 worth of raw materials. Increases Giapetto’s variable labor/overhead costs by $10. Requires 1 hour of finishing labor. Requires 1 hour of carpentry labor.
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3/1 A simple example Each week Giapetto can obtain: All needed raw material. Only 100 finishing hours. Only 80 carpentry hours. Also: Demand for the trains is unlimited. At most 40 soldiers are bought each week. Giapetto’s objective is to maximize weekly profit (revenues - expenses). Formulate a mathematical model that can be used to maximize the weekly profit.
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4/1 A simple example Decision variables : x 1 = number of soldiers produced each week x 2 = number of trains produced each week Objective function : Note that in this example fixed costs do not depend upon the the values of x 1 or x 2 . Weekly profit = weekly revenue - weekly raw material costs - the weekly variable costs = (27 x 1 + 21 x 2 ) - (10 x 1 + 9 x 2 ) - (14 x 1 + 10 x 2 ) = 3 x 1 + 2 x 2
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5/1 A simple example Thus, Giapetto’s objective is to choose x 1 and x 2 to maximize 3 x 1 + 2 x 2 . We use the variable z to denote the objective function value of any LP. Giapetto’s objective function is: Maximize z = 3 x 1 + 2 x 2 “Maximize” will be abbreviated by max and “minimize” by min . The coefficient of an objective function variable is called an objective function coefficient.
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6/1 A simple example Constraints : As x 1 and x 2 increase, Giapetto’s objective function grows larger. However, the values of x 1 and x 2 are limited by the following three restrictions (called constraints): Constraint 1 : Each week, no more than 100 hours of finishing time may be used. Constraint 2 : Each week, no more than 80 hours of carpentry time may be used. Constraint 3 : Because of limited demand, at most 40 soldiers should be produced. These three constraints can be expressed mathematically by the following equations: Constraint 1 : 2 x 1 + x 2 100 Constraint 2 : x 1 + x 2 80 Constraint 3 : x 1 40
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7/1 A simple example The coefficients of the constraints are often called the technological coefficients . The number on the right-hand side of the constraint is called the constraint’s right-hand side (or rhs). Sign Restrictions To complete the formulation of a linear programming problem, the following question must be answered for each decision variable: Can the decision variable only assume nonnegative values, or is the decision variable allowed to assume both positive and negative values?
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