Simplex Method - Management Science - Lecture Slides, Slides of Business Management and Analysis

Download Simplex Method - Management Science - Lecture Slides and more Slides Business Management and Analysis in PDF only on Docsity!

Simplex Method

Overview of the Simplex Method

• Advantages and Characteristics
  • More realistic approach as it is not limited to

problems with two decision variables

• Systematically examines basic feasible solutions for an

optimal solution.

• Based on the solutions of linear equations (equalities)

using slack variables to achieve equality.

• Rule
  • Linear programming models have fewer equations

than variables; unless the number of equations equals

the number of variables, a unique solution cannot be

found.

High-tech Problem (Simplex method):

Solution:

a) Formulation Of Linear Programming Problem:

Decision Variable: let there be two decision variables x 1 and

x 2 , such that

x 1 = number of Deskpros to be produced and

x 2 = number of Portables to be produced

Objective Function:

The objective of the producer is to determine the number

of Deskpros and Portables to be produced, in order to

maximize the total profit, i. e.,

Z = 50x 1 + 40x 2

Constraints:

First Constraint: 3x 1 + 5x 2 <= 150 (Assembly Time)

Second Constraint: x 2 <= 20 (Portable Display)

Third constraint: 8x 1 + 5x 2 <= 300 (Warehouse Capacity)

Non - Negativity Constraint: Since the production process can never be negative, we must have x1 ≥ 0 and x2 ≥ 0.

Re-Write the Model

Max Z = 50x 1 + 40x 2 + 0s 1 + 0s 2 + 0s (^3)

Subject to the constraints,

3x 1 + 5x 2 + s 1 = 150

x 2 + s 2 = 20

8x 1 + 5x 2 + s 3 = 300

And x 1 ,x 2 , s 1 , s 2 ands 3 ≥ 0.

Design the initial Feasible Solution :

An initial basic feasible solution is obtained by putting

x 1 = x 2 = 0 (non –basic variables)

Thus we get,

s 1 = 150 , s 2 = 20, s 3 = 300 (basic variable)

Developing the Initial Simplex Tableau

• Notation used in the simplex tableau:

Setup the initial simplex tableau :

• For computational efficiency and simplicity, initial

basic feasible solution, the constraints of the standard Linear programming problem as well as the objective function can be displayed in the tabular

form, called Simplex Tableau.

***** Incoming Column**

**** Outgoing Row**

*** Key Element (Pivot Element)**

Compute the new Pivot row values by using the formula

New Table Pivot row values = (Old Table Pivot row values) / Pivot Element

= 1/8 (8 5 0 0 1 300) = (1 5/8 0 0 1/8 300/8)

Computing all other row values using the formula New Table Row values = Old row values ― Corresponding Coefficient in Key Column

• New table key row value New First Row Values = Old First Row Values ― 3 * New Table Key row values = (3 5 1 0 0 150) ― (3 15/8 0 0 3/8 225/2) = (0 25/8 1 0 -3/8 75/2) New Second Row Values = Old Second Row Values ― 0 x New Table Key row values = (0 1 0 1 0 20) ― 0*(1 5/8 0 0 1/8 300/8) = (0 1 0 1 0 20)

Compute the new table values by using the formula

New Table Key row values = Old Table Key row values / Key Element New Table Key row values = 8/25(0 25/8 1 0 -3/8 75/2) = (0 1 8/25 0 -3/25 12) Computing all other row values using the formula New Table Row values = Old row values ― Corresponding Coefficient in Key Column * New table key row value New Second Row Values = Old Second Row Values ― 1 x Corresponding New Table Key row values. = (0 1 0 1 0 20) ― (0 1 8/25 0 -3/25 12) = (0 0 -8/25 1 3/25 8) New Third Row Values =Old Third Row Values ― 5/8 x Corresponding New Table Key row values = (1 5/8 0 0 1/8 75/2) ― (0 5/8 1/5 0 -3/40 15/2) = (1 0 -1/5 0 1/5 30)

x 1 x 2 s 1 s 2 s (^3)

Basis (^) Cj^ b (= X^ B) R.H.S CB

x 2 40 0 1 8/25^0 -3/25 b 1 = 12

s 2^0 0 0 -8/25^1 3/25^ b^2 =

x 1 50 1 0 -1/5^0 1/5 b 3 = 30

Zj = Σ C (^) Bj aij = 0 50 40 14/5 0 26/

Z =ΣCj X (^) B =1240+30 = Contribution to profit C** (^) j -Zj

0 0 -14/5 0 -26/

New Table (Final Solution):

No more improvement is possible as all the C (^) j -Zj a re either 0 or negative. So we stop at this point.