a prototype example: the galaxy linear programming model
DESCRIPTION
A Prototype Example: The Galaxy Linear Programming Model. Max 8X 1 + 5X 2 (Weekly profit) subject to 2X 1 + 1X 2 £ 1000 (Plastic) 3X 1 + 4X 2 £ 2400 (Production Time) X 1 + X 2 £ 700 (Total production) X 1 - X 2 £ 350 (Mix) - PowerPoint PPT PresentationTRANSCRIPT
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Max 8X1 + 5X2 (Weekly profit)subject to2X1 + 1X2 1000 (Plastic)
3X1 + 4X2 2400 (Production Time)
X1 + X2 700 (Total production)
X1 - X2 350 (Mix)
Xj> = 0, j = 1,2 (Nonnegativity)
A Prototype Example: A Prototype Example: The Galaxy Linear Programming ModelThe Galaxy Linear Programming Model
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The Graphical Analysis of Linear The Graphical Analysis of Linear ProgrammingProgramming
The set of all points that satisfy all the constraints of the model is called
a
FEASIBLE REGIONFEASIBLE REGION
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Using a graphical presentation
we can represent all the constraints,
the objective function, and the three
types of feasible points.
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The non-negativity constraints
X2
X1
Graphical Analysis – the Feasible RegionGraphical Analysis – the Feasible Region
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1000
500
Feasible
X2
Infeasible
Production Time3X1+4X2 2400
Total production constraint: X1+X2 700 (redundant)
500
700
The Plastic constraint2X1+X2 1000
X1
700
Graphical Analysis – the Feasible RegionGraphical Analysis – the Feasible Region
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1000
500
Feasible
X2
Infeasible
Production Time3X1+4X22400
Total production constraint: X1+X2 700 (redundant)
500
700
Production mix constraint:X1-X2 350
The Plastic constraint2X1+X2 1000
X1
700
Graphical Analysis – the Feasible RegionGraphical Analysis – the Feasible Region
• There are three types of feasible pointsInterior points. Boundary points. Extreme points.
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Solving Graphically for an Solving Graphically for an Optimal SolutionOptimal Solution
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The search for an optimal solutionThe search for an optimal solution
Start at some arbitrary profit, say profit = $2,000...Then increase the profit, if possible...
...and continue until it becomes infeasible
Profit =$4360
500
700
1000
500
X2
X1
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Summary of the optimal solution Summary of the optimal solution
Space Rays = 320 dozen Zappers = 360 dozen Profit = $4360
– This solution utilizes all the plastic and all the production hours.
– Total production is only 680 (not 700).
– Space Rays production exceeds Zappers production by only 40
dozens.
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– If a linear programming problem has an optimal solution, an extreme point is optimal.
Main Result: Extreme points and optimal Main Result: Extreme points and optimal solutionssolutions
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• Linear programming software packages solve large linear models i.e. many decision variables and many constraints.
• Graphical method is limited to 2-decision variable LP problems, however, LP software packages use the Main Result of graphical method, called the Simplex algorithm.
• The input to any package includes:– The objective function criterion (Max or Min).– The type of each constraint: .– The actual coefficients for the problem.
Computer Solution of Linear Programs With Computer Solution of Linear Programs With Any Number of Decision VariablesAny Number of Decision Variables
, ,