Linear Programming Part 1

Linear Programming Part 1 - Linear Programming Part I DS...

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Linear Programming Part I DS 412 Katy Azoury

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Linear Programming A Mathematical model for formulating and solving managerial decision making problems Method was developed in early 1950’s It is very widely used today. In general, linear programming (LP) provides the optimal solution ( best decisions ) to a wide range of decision problems. The approach can handle very large problems. LP does not model uncertainty. There are many specialized software packages that solve LP models.
Characteristics of LP models Decision Variables : These represent the alternative set of decisions to choose from. They are the unknowns variables in the LP model Objective: The goal that we want to optimize (maximized or minimized). Objectives are usually profits or costs. Constraints: represent restrictions or limits on our decisions. describe the environment of a linear programming application. quantify how our resources are limited or how our requirements must be met. Linearity: the objective and the constraints must be expressed as linear expressions and linear inequalities in terms of the decision

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Linear/Non-linear Equations/Inequalities Linear examples: Linear expression: 3X + 2Y Linear equation: 7X + 5Y + 10Z = 220 Linear inequality: 21X + 12Y – 3Z <= 342 Non Linear examples: X/Y + Z X2 + 7Y = 12 X3 -1/Y + Z1/2 >= 300 Linear expressions, equations and inequalities are allowed in LP models. Non linear expressions, equations and inequalities are not
Standard Model Structure for LP models Objective Function : A linear expression in terms of the decision variables . Subject to: Constraints Linear expression Left hand side of the constraint or or = Number Right hand side of the constraint

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Product Mix Example A furniture company makes chairs desks. Each of these two products requires labor hours in two departments; fabrication and assembly. Demand for these products is strong, so they can sell all that can produce. A chair requires 5 hours in fabrication and 4 hours in assembly. A desk requires 10 hours in fabrication and 4 hours in assembly. There are 300 hours available in the fabrication department and 200 hours in assembly. Resource costs are \$18 for one hour of fabrication and \$15 for one hour of assembly.
Profit calculations Chair Desk Revenue \$180 per chair 280 per desk Fab. Cost 5*18 = \$90 10*18 = \$180 Assembly cost 4*15 = \$60 4*15 = \$60 Total cost \$150 \$ 240 Profit per unit \$30 \$ 40

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Model Formulation Define the decision variables: Let X = number of chairs produced Let Y = number of tables produced Objective Function : Maximize Profit = 30X + 40Y Subject to: Constraints 5X + 10Y≤ 300 fabrication 4X + 4Y≤ 200 assembly X 15 client demand
Graphical Solution Two variable problems can be solved graphically.

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Linear Programming Part 1 - Linear Programming Part I DS...

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