Linear Programming

Topics: Optimization, Constraint, Inequality Pages: 5 (887 words) Published: August 30, 2013
TOPIC – LINEAR PROGRAMMING

Linear Programming is a mathematical procedure for determining optimal allocation of scarce resources.  
Requirements of Linear Programming
•        all problems seek to maximize or minimize some quantity •        The presence of restrictions or constraints •        There must be alternative courses of action •        The objective and constraints in linear programming must be expressed in terms of linear equations or inequalities

 
Objective Function         it maps and translates the input domain (the feasible region) into output range, with the two-end values called the maximum and minimum values  

Restriction Constraints it limits the degree to which we can pursue our objective  
Decision Variables         represents choices available to the decision maker in terms of amount of either inputs or outputs  
Parameters these are the fixed values in which the model is solved  
Basic Assumption of Linear Programming        
1. Certainty- figures or number in the objective and constraints are known with certainty and do not vary 1. Proportionality - for example 1:2 is equivalent to 5:10 1. Additivity - the total of all the activities equals the sum of the individual activities. 1. Divisibility - solutions to the LP problems may not be necessary in whole (integers) numbers, hence, divisible and can assume any fractional value 1. Non-negativity - cannot use or produce negative physical quantities.  

Procedures in Graphical Solution        
1. Set up the objective function and constraints in mathematical format. 1. Plot the constraints
1. Identify the feasible solution space
1. Plot the objective function
1. Determine the optimum solution                                                                  
Sample Problem Set        
 
Department        Tables ( T )         Chairs ( C )         Available hours Carpentry         4         3 240 Painting         2         1         100  

Profit Amount : Table P7.00 Chair P5.00
 
LP Model
 
Objective :
Max Z = 7 T + 5 C
 
Subject to :
Carpentry > 4 T + 3 C ≤ 240
Painting > 2 T + C ≤ 100
Non-negativity > T, C > 0
 
Solution :
1.        Mathematical
2.        Graphical
 
Mathematical
 
4 T + 3 C = 240
2 T + C = 100
 
4 T + 3 C = 240
3 ( 2 T + C = 100 ) to eliminate C, multiply by 3 so C with both be 3C
4 T + 3 C = 240
6 T + 3 C = 300 next, subtract 2nd equation to 1st equation  
4 T + 3 C = 240
- 6 T - 3 C = - 300
- 2 T = - 60 next, subtract 2nd equation to 1st equation
T = - 60 / - 2

T = 30
 
Then substitute this value to any of the equations:

4 T + 3 C = 240
4 (30) + 3 C = 240
120 + 3 C = 240
3 C = 240 - 120
C = 120 / 3
C = 40
 
Then substitute the T and C values to the Objective Function to get the Minimum Cost  
Max Z = 7 T + 5 C
= 7 (30) + 5 (40)
= 210 + 200
Z = P410

Remember :
 
In maximization, sign of inequality is ≤ (less than or equal to). In minimization, sign of inequality is ≥ (more than or equal to).  
In maximization, solution space is the area shaded below the constraints. In minimization, solution space is the area shaded above the constraints....
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