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**Dr. Ayham Jaaron First semester 2013/2014 August 2013**

Operations Research I Chapter 02 (continued) Modeling with Linear Programming Dr. Ayham Jaaron First semester 2013/2014 August 2013

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**Model formulation Example for practice**

Farmer Jones decided to supplement his income by baking and selling two types of cakes, chocolate and vanilla. Each chocolate cake sold gives a profit of $3, and the profit on each vanilla cake sold is $5. each chocolate cake requires 20 minutes of baking time and uses 4 eggs and 4 pounds of flour, while each vanilla cake requires 40 minutes of baking time and uses 2 eggs and 5 pounds of flour. If farmer Jones has available only 260 minutes of baking time, 32 eggs, and 40 pounds of flour, how many of each type of cake should be baked in order to maximize farmer Jones’ profit?

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Problem No.4 (book page 26)

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Problem 14 (Book page 21) Construct the LP model that represents the problem below? Maximize steam generated!!!

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**Operations Research I Chapter 02 (continued) Graphical LP Solution**

Dr. Ayham Jaaron

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**Graphical LP Solution (2 variables)**

The graphical procedure includes two steps: Determination of the feasible solution space Determination of the Optimum solution from among all the feasible points in the solution space.

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**Graphical solution types**

Maximization Problems Minimization problems We shall try with Maximization problems first

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**Example 1: The Reddy Mikks Company**

Maximize Z= 5 X1 + 4 X2 Let’s try now !

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**Example 1: The Reddy Mikks Company**

Maximize Z= 5 X1 + 4 X2

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**Example 1: The corner points technique**

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**Another Technique: using arbitrary values**

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**Example (2) on Graphical Method**

Resolve using the Graphical Method for the following problem: Maximize Z = 3x + 2y subject to: x + y ≤ 18 2x + 3y ≤ 42 3x + y ≤ 24 x ≥ 0 , y ≥ 0

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Example (2)...Continued Initially we draw the coordinate system correlating to an axis the variable x, and the other axis to variable y, as we can see in the following figures.

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Example (2)...continued

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Example (2)...continued we proceed to determine the extreme points in the feasible region, candidates to optimal solutions, that are the O-F-H-G-C points figure's. Finally, we evaluate the objective function ( 3x + 2y ) at those points, which result is picked up in the following board. As G point provides the bigger value to the objective Z, such point constitutes the optimal solution, we will indicate x = 3; y = 12, with optimal value Z = 33.

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Example (2)...continued

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**Example (3): Reddy Mikks Company problem **

Graphical Solution of Maximization Model (1 of 12) X2 is mugs Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 X1 is bowls Figure 2.2 Coordinates for Graphical Analysis

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**Graphical Solution of Maximization Model (2 of 12)**

Labor Constraint Graphical Solution of Maximization Model (2 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure 2.3 Graph of Labor Constraint

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**Graphical Solution of Maximization Model (3 of 12)**

Labor Constraint Area Graphical Solution of Maximization Model (3 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure 2.4 Labor Constraint Area

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**Graphical Solution of Maximization Model (4 of 12)**

Clay Constraint Area Graphical Solution of Maximization Model (4 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure 2.5 Clay Constraint Area

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**Graphical Solution of Maximization Model (5 of 12)**

Both Constraints Graphical Solution of Maximization Model (5 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure 2.6 Graph of Both Model Constraints

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**Feasible Solution Area **

Graphical Solution of Maximization Model (6 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure Feasible Solution Area

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**Objective Function Solution = $800 **

Graphical Solution of Maximization Model (7 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure 2.8 Objection Function Line for Z = $800

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**Alternative Objective Function Solution Lines **

Graphical Solution of Maximization Model (8 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure 2.9 Alternative Objective Function Lines

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**Graphical Solution of Maximization Model (9 of 12)**

Optimal Solution Graphical Solution of Maximization Model (9 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure Identification of Optimal Solution Point

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**Optimal Solution Coordinates **

Graphical Solution of Maximization Model (10 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure 2.11 Optimal Solution Coordinates

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**Extreme (Corner) Point Solutions **

Graphical Solution of Maximization Model (11 of 12) Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure Solutions at All Corner Points

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**Optimal Solution for New Objective Function **

Graphical Solution of Maximization Model (12 of 12) Maximize Z = $70x1 + $20x2 subject to: 1x1 + 2x2 40 4x1 + 3x2 120 x1, x2 0 Figure 2.13 Optimal Solution with Z = 70x1 + 20x2

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**Graphical Solution for Minimization Problems**

To apply the Graphical method to solve a minimization LP Model, consider the following problem Minimize: Z= 0.3X X2 Subject to: x1+x2 ≥ 800 (0, 800), (800,0) 0.21x x2 ≤ (0,0) , (100,70) 0.03x1-0.01x2 ≥ (0,0), (25,75) x1,x2 ≥ 0

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**Example (2) minimization problem using Graphical solution**

Solve the following LP model using graphical solution: Minimize: z = 8X1 + 6X2 Subject to: 4X1 + 2X2 >=20 -6X1+4X2 <=12 X1 + X2 >=6 X1, X2>=0

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Home work No.1 Home work Number 1 is due to be submitted on Wednesday 11th September No late submissions will be accepted under any circumstances. Page 15: Problems 1, 2 Page 20: Problems 4, 5, 6

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