Linear programming Simplex method

Group Members Hania Ahmed Misbah Shuja Ansari Syed Ikram akbar

Linear Programming  Linear programming (LP, or linear optimization) is a mathematical method for determining a way to achieve the best outcome Mathematical technique helps managers in their planning and decision making Linear programming problem is one in which we are to find the maximum or minimum value of a linear expression ax + by + cz ………………………………………………………………………………………. ….. ………………………………… …………………………………………………………………….0

Examples of Linear programming Problems Selection of product mix in a factory The Airline industry uses it in order to optimize profits and minimize expenses in their business Farmers uses it to increase the revenue of their operations

Simplex Method In mathematical optimization theory, the simplex method was created by the American George dantzig in 1947 The Simplex Algorithm is a method of solving linear programming problems. It is used to reach a goal while having constraints The journal computing in science and enigneering listed it as one of the top 10 algorithms of 20th century

Simplex Method Typical uses of the simplex algorithm is to find the most effective method to perform a task at the lowest cost (the goal). Many manufacturing industries rely on simplex method such as for Transportation systems

Simplex Method It provides an efficient procedure for solving large problems Easily programmed on a computer and Easy to use

Objective Function The function to be maximized or minimized subject to constraints is called the objective function : 3x + 4y + 5z Constraints are limitations for example lack of resources 3x + 4y + 5z ≤ 30 In case of profit maximization, it’s called profit function.In case of cost, it’s called cost function

3x + 4y + 5z ≤ 30 The variables x, y, z, are called the decision variables. They represent quanitities to be determined Collection of values of x, y, z gives the optimal value that constitutes an optimal solution It is a feasible solution with the largest objective function value

Slack Variable The dummy variable added to inequality ≤ constraint to transform it into equality Either it can be positive or zero Generally if the constraint is in the form ≥ then the variable subtracted is called surplus variable

Example Constraint inequality in a given situation is: 4x + 3y ≤ 30 to convert inequality into equation, add slack variable 4x + 3y + S = 30 Surplus variable example: 3x + 2y ≥ 25 Subtract surplus variable and add artificial variable 3x + 2y – S1 + A1 = 25

Basic Rules of Simplex The right hand side value of constraints is called constants or bi 4x + 3y ≤ 25 There should be only constants on RHS All the values on RHS of constraints should be non-negative In case it is negative, make it non-negative by multiplying both sides by (-1)

Graphical Concepts Graphical concepts describe how to plot the feasible set and objective function Two decision variables are plotted on a graph. We then plot all the constraints on the same graph When these lines intersect, the feasible region is identified.

Graphical Analysis

Some related concepts A basic solution of a linear programming problem in standard form is a solution of the constraint equations in which at most m variables are nonzero–– where m is equal to the number of slack variables the variables that are nonzero are called basic variables. A basic solution for which all variables are nonnegative is called a basic feasible solution

Simplex Method Steps Add slack variables to change the constraints into equations and write all variables to the left of the equal sign and constants to the right. Write the objective function with all nonzero terms to the left of the equal sign and zero to the right. The variable to be maximized must be positive.

Simplex Method Steps Set up the initial simplex tableau by creating an augmented matrix from the equations, placing the equation for the objective function last ( in the bottom row) Determine a pivot element and use matrix row operations to convert the column containing the pivot element into a unit column

Pivot Element The entering variable corresponds to the smallest (the most negative) entry in the bottom row of the tableau. The departing variable corresponds to the smallest nonnegative ratio of , in the column determined by the entering variable. The entry in the simplex tableau in the entering variable’s column and the departing variable’s row is called the pivot.

Pivoting To form the improved solution, we apply Gauss-Jordan elimination to the column that contains the pivot, This process is called pivoting.

Simplex Method Steps If negative elements still exist in the bottom row, repeat Step 4. If all elements in the bottom row are positive, the process has been completed. When the final matrix has been obtained, determine the final basic solution. This will give the maximum value for the objective function and the values of the variables where this maximum occurs.

Example The Ace Novelty Company has determined that the profits are $6, $5, and $4 for each type-A, type-B, and type-C souvenir that it plans to produce. To manufacture a type-A souvenir requires 2 minutes on machine I, 1 minute on machine II, and 2 minutes on machine III. A type-B souvenir requires 1 minute on Machine I, 3 minutes on machine II, and 1 minute on machine III. A type-C souvenir requires 1 minutes on machine I and 2 minutes on each of machines II and III. There are 3 hours available on machine I, 5 hours available on machine II, and 4 hours available on machine III for manufacturing these souvenirs each day. How many souvenirs of each type should Ace Novelty make per day in order to maximize its profit?

Solution The objective function is P = 6x + 5y + 4z, which is to be maximized The constraints are 2x + y + z ≤ 180 x + 3y + 2z ≤ 300 2x + y + 2z ≤ 240 X, y, z ≥ 0,

Solution Insert slack variables to change inequalities into equations; rewrite objective function 2x + y + z + s = 180 x + 3y + 2z + t = 300 2x + y + 2z + u = 240 – 6x – 5y – 4z + P = 0

Basic s t u

Select the pivot element Select the column with the most negative indicator: column 1 in this tableau. Divide each constant to the right of the bar by the corresponding (nonzero) element in the pivot column. 180/2 = 90 select the smallest quotient 300/1 = 300 240/2 = 120 The pivot element is the intersection of the column with the most negative indicator and the row with the smallest quotient.

Basic x t u Basic x t u

Solution Determine a new pivot element Basis x y u

Basis x y u

Answer check P = 6x + 5y + 4z P = 6(48) + 5(84) + 4(0) 708 = 288 + 420