LP3:The Galaxy Industries Production Problem

8/12/2019 LP3:The Galaxy Industries Production Problem

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The Galaxy Industries Production Problem

A Prototype Example

Galaxy manufactures two toy doll models:

Space Ray.

Zapper.

Resources are limited to

1000 pounds of special plastic.

40 hours of production time per week.


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Marketing requirement

Total production cannot exceed 700 dozens.

Number of dozens of Space Rays cannot exceed number of

dozens of Zappers by more than 350.

Technological input

Space Rays requires 2 pounds of plastic and

3 minutes of labor per dozen.

Zappers requires 1 pound of plastic and

4 minutes of labor per dozen.


A Prototype Example


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The current production plan calls for:

Producing as much as possible of the more profitable product,Space Ray ($8 profit per dozen).

Use resources left over to produce Zappers ($5 profit

per dozen), while remaining within the marketing guidelines.


A Prototype Example

X1 = No of dozens to be produced of Space RayX2 = No of dozens to be produced of Zapper


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Max 8X1+ 5X2 (Weekly profit)

subject to

2X1+ 1X21000 (Plastic)

3X1+ 4X22400 (Production Time)

X1+ X2

700 (Total production)X1 - X2 350 (Mix)

Xj> = 0, j = 1,2 (Nonnegativity)

The Galaxy Linear Programming Model


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1000

500

Feasible

X2

Infeasible

ProductionTime3X1+4X2 2400

Total production constraint:

X1+X2700 (redundant)500

700

The Plastic constraint

2X1+X2 1000

X1

700

Graphical Analysisthe Feasible Region


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1000

500

Feasible

X2

Infeasible

ProductionTime3X1+4X2 2400

Total production constraint:

X1+X2 700 (redundant)500

700

Production mixconstraint:

X1-X2 350

The Plastic constraint

2X1+X2 1000

X1

700

Graphical Analysisthe Feasible Region


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The search for an optimal solution

Profit =$4360

500

700

1000

500

X2

X1


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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).

Zappers production exceeds Space Rays production by 40

dozens.


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If a linear programming problem has an optimalsolution, an extreme point is optimal.

Extreme points and optimal solutions


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For multiple optimal solutions to exist, the objectivefunction must be parallel to one of the constraints

Multiple optimal solutions


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1

No point, simultaneously,

lies both above line and

below lines and

.

1

2 32

3

Infeasible Model


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Unbounded solution

LP3:The Galaxy Industries Production Problem

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