# 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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