Mathematical Problem (Simplex Model)

Problem 3: Dell Inc. manufactures two sizes of baseball- little league and major league. The company earns a profit of $2 of little league baseball and $3 per of major league baseball. Each product is assembled and packaged. There is a maximum of 1,800 hours available between the assembling and packaging department during a given time period. It takes 9 minutes to assemble a box of little league baseball and 15 minutes to assemble a box of major league baseball. A box of little league requires eleven minutes packaging whereas 5 minutes to major league. Dell Inc. seeks 2 combination of little league baseball and major league baseball that will maximize total profit within the available assembling and packaging time.

Requirement: Formulate this problem with simplex model and solve it.

Key column / Key Row / Key number

Key column: In a simplex table the column which contains the most positive number in the objective row is called Key column or pivot column.

Key Row: In a simplex table the raw which contains the lowest displacement ratio is called key row or pivot row.

Key number: In a simplex table the number which lies at the intersection of the key row and key column is called key number.

Surplus variables

Surplus variables

In a linear programming model when a structural constraint is in the “Greater  than or equal to” form a non-negative variable is deducted to the left hand side to convert inequality into equation, then these variables are called as slack variables.

For example: 3X + 5Y ≤ 100

Or, 3X - S₁ + 5Y - S₂ = 100.

Here S₁ and S₂ are Surplus variables.

 

Slack Variables

Slack Variables

In a linear programming model when a structural constraint is in the “less than or equal to” form a non-negative variable is added to the left hand side to convert inequality into equation, then these variables are called as slack variables.

For example: 3X + 5Y ≤ 100

Or, 3X + S₁ + 5Y + S₂ = 100.

Here S₁ and S₂ are slack variables.

Simplex Method

Simplex Method

A methodology designed to systematically solve large scale linear programming problems. This method is an algebraic approach based equality relationship. Yet linear programs typically involve inequality. To use the simplex method, the decision makers first must convert each inequalities restriction into equality through adding slack variables of deducting surplus variables. Simplex method was first introduced by G.B Dantzig and his associates in 1947 in USA among the department of airforce.