OMIS 327 Lecture 4. LP Graphical Analysis

OMIS 327 Lecture 4. LP Graphical Analysis - OMIS 327...

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1 OMIS 327 Lecture 4: LP Graphical Analysis Instructor: Dr. Lee, Jung Young
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2 Today’s Learning Goals Graphical Solution to an LP Problem Isoprofit Line Solution Method Corner Point Solution Method Slack and Surplus
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3 Flair Furniture Company Maximize profit = $70 T + $50 C subject to 4 T + 3 C ≤ 240 (carpentry constraint) 2 T + 1 C ≤ 100 (painting and varnishing con T ≥ 0 (nonnegativity constraint) C ≥ 0 (nonnegativity constraint) T = number of tables to be produced per week. C = number of chairs to be produced per week.
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4 Graphical Representation of a Constraint The first step in solving the problem is to identify a set or region of feasible solutions. To do this we plot each constraint equation on a graph. We start by graphing the equality portion of the constraint equations: 4 T + 3 C = 240 We solve for the axis intercepts and draw the line.
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5 Graphical Representation of a Constraint 100 – 80 – 60 – 40 – 20 – C | | | | | | | | | | | | 0 20 40 60 80 100 T Number of Chairs Number of Tables This Axis Represents the Constraint T ≥ 0 This Axis Represents the Constraint C ≥ 0 Figure 7.1 Quadrant Containing All Positive Values
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6 Graphical Representation of a Constraint When Flair produces no tables, the carpentry constraint is: 4(0) + 3 C = 240 3 C = 240 C = 80 Similarly for no chairs: 4 T + 3(0) = 240 4 T = 240 T = 60 This line is shown on the following graph:
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7 Graphical Representation of a Constraint 100 – 80 – 60 – 40 – 20 – C | | | | | | | | | | | | 0 20 40 60 80 100 T Number of Chairs Number of Tables ( T = 0, C = 80) Figure 7.2 ( T = 60, C = 0) Graph of carpentry constraint equation
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8 Graphical Representation of a Constraint 100 – 80 – 60 – 40 – 20 – C | | | | | | | | | | | | 0 20 40 60 80 100 T Number of Chairs Number of Tables Figure 7.3 Any point on or below the constraint plot will not violate the restriction. Any point above the plot will violate the restriction. (30, 40) (30, 20) (70, 40) Region that Satisfies the Carpentry Constraint
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9 Graphical Representation of a Constraint The point (30, 40) lies on the plot and exactly satisfies the constraint 4(30) + 3(40) = 240. The point (30, 20) lies below the plot and satisfies the constraint 4(30) + 3(20) = 180.
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