1. LINEAR PROGRAMMING

2. Linear Programming
Technique for economic allocation of scarce resources like man, machine, material etc to several competing activities such as products, services etc on the basis of given criterion of optimality.
Linear relationships used in mathematical modeling
LINEAR PROGRAMMING

3. General Structure of LP
LP model, as in any OR model, includes three basic elements:
Decision variables that we seek to determine
Objective (goal) that we aim to achieve
Constraints that we need to satisfy
Maximize or Minimize a Linear objective subject to linear equalities or non-equalities (constraints).
LINEAR PROGRAMMING

4. Terminology
Objective Function - Value measure used to rank alternatives, Seek to maximize or minimize this objective, examples: maximize NPV, minimize cost. Example 3x + 5y , x & y are decision variables.
Decision Variables - Quantities you can control to improve your objective which should completely describe the set of decisions to be made.
Constraints - Limitations on the values of the decision variables. Example 2x + 3y ≤24, x ≥0, y ≥0
LINEAR PROGRAMMING

5. Mathematical Model Optimize (max or min) Z
Z = measure of performance variable
c1, c2, …cn represent contribution of a unit of the respective variable to performance of z
x1, x2, …. xn are decision variables
subject to linear , non-negative constraints.
LINEAR PROGRAMMING

6. Algebraic Formulation
Optimize (max or min) the objective function
subject to linear constraints
i = 1,2,…m constraints
non-negativity constraints
LINEAR PROGRAMMING

7. Guidelines on LP model Formulation
Step 1 – Identify decision variables.
Express each constraint in words. Identify the equality condition
Express objective function verbally
Identify the decision variables verbally
Step 2 – Identify the problem data
Step 3 – Formulate the constraints
Convert the verbal constraint equation in mathematical form.
Step 4 – Formulate the objective function in mathematical form
LINEAR PROGRAMMING

8. Example of LP Model
Reddy Mikks produces both interior and exterior paints from two raw materials, M1 and M2. The following table provides the basic data of the problem
A market survey restricts the maximum demand of interior paint to 2 tons. Additionally, the daily demand of interior paint cannot exceed that of exterior paint by more than 1 ton. Reddy Mikks wants to determine the optimum (best) product mix of interior and exterior paints that maximize daily profit.
LINEAR PROGRAMMING

9. Reddy Mikks – Objective Function
We need to determine the amount of interior and exterior paints to be produced.
The decision variables in this problem are defined as
x1 – Exterior paint produced per day
x2 – Interior paint produced per day
The objective is to maximize profit. Contribution of exterior is $5000/Ton and interior paint is $4000/Ton.
Objective function will be
Maximize Z (profit) = 5x1 + 4x2
LINEAR PROGRAMMING

10. Reddy Mikks - Constraints
The first set of constraints is the raw material
Usage of raw material by both paints must be less than available raw materials.
Usage of raw material M1 is 6x1+4x2 for exterior and interior paints.
Similarly for M2 it is 1x1+2x2 for exterior and interior paints.
Available raw material is 24 Tons of M1 and 6 Tons of M2
Hence constraint equations are
6x1 + 4x2 24 (Raw material M1)
x1 + 2x2 6 (Raw material M2)
LINEAR PROGRAMMING

11. Reddy Mikks - Constraints
The second set of constraints is the demand.
Maximum daily demand of interior paint is 2 T
And the demand for interior paint over exterior paint cannot exceed 1 T.
Expressed mathematically these are
x2  2
x2 – x1  1
Finally the non-negativity constraint gives us
x1, x2 0
LINEAR PROGRAMMING

12. The Complete Model Objective Function Constraints
Maximize Z (profit) = 5x1 + 4x2
Constraints
6x1 + 4x2 24 (Raw material M1)
x1 + 2x2 6 (Raw material M2)
x2  2
-x1 + x2  1
x1, x2 0
LINEAR PROGRAMMING

13. LP Assumptions
Proportionality requires the contribution of each decision variable in both the objective function and the constraints to be directly proportional to the value of the variable. In the RM example, if quantity discounts are offered then the revenues will not be proportional to sales.
Additivity stipulates that the total contribution of all the variables in the objective function and in the constraints to be the direct sum of individual contribution of each variable.
Certainty stipulates that each linear coefficient of the objective function and constraints is known.
Divisibility stipulates that the variable is assumed to take fractional values.
LINEAR PROGRAMMING

14. One More example
A tape recorder company manufactures models A, B & C, which have profit contribution per unit of Rs 15, Rs 40 and Rs 60 respectively. The wekly minimum production requirements are 25 units of model A, 130 units for model B, and 55 units of model C. A dozen units of model A requires 4 hours for manufacturing, 3 hours for assembling and 1 hour for packaging. The corresponding figures for dozen units of B & C are (2.5, 4,2) and (6,9,4) respectively. During the forthcoming week the company has available 130 hours of manufacturing, 170 hours of assembling and 52 hours of packaging. Formulate the LP model.
LINEAR PROGRAMMING

15. Solution
Ten minutes to solve.
LINEAR PROGRAMMING