1. 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

2. 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?

3. Problem No.4 (book page 26)

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

5. Operations Research I Chapter 02 (continued) Graphical LP Solution
Dr. Ayham Jaaron

6. 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.

7. Graphical solution types
Maximization Problems
Minimization problems
We shall try with Maximization problems first

8. Example 1: The Reddy Mikks Company
Maximize Z= 5 X1 + 4 X2
Let’s try now !

9. Example 1: The Reddy Mikks Company
Maximize Z= 5 X1 + 4 X2

10. Example 1: The corner points technique

11. Another Technique: using arbitrary values

12. 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

13. 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.

14. Example (2)...continued

15. 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.

16. Example (2)...continued

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

31. 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

32. 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