MBA Semester linear analysis and programming function

Linear programming

1. Linear Programming Vikash Singh Senior Data Scientist, MBA, BSc (Statistics), UGC Net 18+ years of experience in DS, ML, AI and Strategy

2. Session 7 Introduction to Linear Programming Problems LPP Formulations Graphical Solution Session 8 Simplex method Training on Excel What we will cover:

3. Linear Programming Problems (LPP): Linear programming or linear optimization is a process which takes into consideration certain linear relationships to obtain the best possible solution to a mathematical model. It is also denoted as LPP. Linear programming is a process that is used to determine the best outcome of a linear function. It is the best method to perform linear optimization by making a few simple assumptions. The linear function is known as the objective function. Real-world relationships can be extremely complicated. However, linear programming can be used to depict such relationships, thus, making it easier to analyze them. LPP

4. Simplex Method is one of the most powerful & popular methods for linear programming. The simplex method is an iterative procedure for getting the most feasible solution. In this method, we keep transforming the value of basic variables to get maximum value for the objective function. Simplex Method

5. Simplex Solution

6. Simplex Solution - Steps 177 < 180 177 + 3 = 180

7. 1. Simplex Table

8. 2. Pivot Column Pivot column is the one with smallest value which is -6 so this becomes my pivot column.

9. 2. Pivot Row Divide the constants by corresponding values of the pivot column. Coefficients and constants

10. Pivot Row Divide the constants by corresponding values of the pivot column. this is smallest, so becomes pivot row. 2 is the pivot cell

11. Multiply the pivot row so that the pivot cell becomes 1 No change anything in 2nd and 3rd row

12. Make the other elements of the pivot column to zero

13. Make the other elements of the pivot column to zero

14. Now which column is the pivot column???? Now this is the new table.

15. we have two negatives, and the largest negative is -2. so that becomes now a pivot column. Now this is the new table.

16. Find the pivot row???

17. Make pivot cell equal to 1

18. Make other cells of pivot row zero

19. Make other cells of pivot row zero Now we have no negative cells in last row

20. Final simplex matrix In a simplex matrix, the variables that have a unit column are called basic variables. A unit column is a column that is all zeros except for a single one. The variables that are not a unit column, i.e. a column of junk, are called non-basic variables.

21. Final simplex matrix In a simplex matrix, the variables that have a unit column are called basic variables. A unit column is a column that is all zeros except for a single one. The variables that are not a unit column, i.e. a column of junk, are called non-basic variables. The non-basic variables act like parameters and thus in the simplex process, we set their values to zero.

22. Final simplex matrix Maximize P: 6x + 5y + 4z : This is happening at x = 48, y = 84, z =0, P = 708

23. Formulation of LPP in MS-Excel Problem: The World Light Company produces two light fixtures (product 1 and 2) that require both metal frame parts and electrical components. Management wants to determine how many units of each product to produce so as to maximize profit. For each unit of the product 1, 1 unit of frame parts and 2 units of electrical components are required. For each unit of product 2, 3 units of frame parts and 2 units of electrical components are required. The company has 200 units of frame parts and 300 units of electrical components. Each unit of product 1 gives a profit of $1, and each unit of product 2, upto 60 units, gives a profit of $2. Any excess over 60 units of product 2 brings no profit, so such an excess has to be ruled out. Formulate the LP model for this problem Use the graphical method to solve this model. What is the resulting total profit? Solve this using MS-Excel solver, Python and R-programing (optional).

24. Problem Definition The WorldLight Company produces two light fixtures (product 1 and 2) that require both metal frame parts and electrical components. Management wants to determine how many units of each product to produce so as to maximize profit. Number of units of Product 1 and Product 2. Let X1, be the number of units of Product 1 X2, be the number of units of Product 2 Identifying the variables

25. Problem Definition Each unit of product 1 gives a profit of $1, and each unit of product 2, upto 60 units, gives a profit of $2. Objective of the problem is to maximize the profit   Max Z = P = 1 * X1 + 2 * X2 5 units of pr 1 = x1 = 5 4 units of pr 2 = x2 = 4 P = 5 * 1 + 4 * 2 = 13dollars Identify the objective of the problem:

26. Problem Definition For each unit of the product 1, 1 unit of frame parts and 2 units of electrical components are required. For each unit of product 2, 3 units of frame parts and 2 units of electrical components are required. The company has 200 units of frame parts and 300 units of electrical components. Any excess over 60 units of product 2 brings no profit, so such an excess has to be ruled out. Constraints: Constraints Product 1 Product 2 Availability Raw material constraint:       Metal frame parts 1 3 200 Electrical components 2 2 300 Production constraint:   <=60  

27. Problem Definition Raw material constraint: For Me tal frame: 1X1 + 3X2 <=200 For Electrical component: 2 X1 + 2X2 <=300 Production constraint: For Product 2: X2<=60 Constraints: Constraints Product 1 Product 2 Availability Raw material constraint:       Metal frame parts 1 3 200 Electrical components 2 2 300 Production constraint:   <=60   Non-negativity condition: X1,X2>=0

28. Problem Definition Objective function: Max Z = 1X1 + 2X2 Subject to the constraint: Raw material constraint: For Me tal frame: 1X1 + 3X2 <=200 For Electrical component: 2 X1 + 2X2 <=300 Production constraint: For Product 2: X2<=60 Non-negativity condition: X1,X2>=0 Problem:

29. Problem Definition – In Excel Problem:

30. Problem Definition – In Excel Problem:

31. Problem Definition – In Excel Solver in MS-Excel:

32. Installing Solver in Windows

33. Installing Solver in Mac OS

34. Problem Definition – In Excel Problem:

35. Problem Definition – In Excel Solver Parameters: Objective function Decision variables Constrains Solving method

36. Problem Definition – In Excel Solver Parameters: Objective function Decision variables Constrains Solving method

37. Problem solving – In Excel Solver in MS-Excel: Objective function Decision variables Constrains Solving method

38. Problem solving – In Excel Problem:

39. Solution– In Excel Problem: Hence the company should produce 1 unit of X1 and 2 units of X2 to make the maximum profit of 180. This is achieved by using 180 units of metal frame and 300 units of electrical component. The numbe r of units of product 2 has not exceeded 60 units as defined in the constraint

40. https://www.cuemath.com/algebra/linear-programming/ https://www.analyticsvidhya.com/blog/2017/02/lintroductory-guide-on-linear-programming-explained-in-simple-english/ https://www.spiceworks.com/tech/it-strategy/articles/linear-programming/#:~:text=Linear%20programming%20(LP)%20uses%20many,restrictions%20of%20supplies%20and%20personnel . https://www.youtube.com/watch?v=rQt_SWrOktg&ab_channel=TimMelvin https://www.youtube.com/watch?v=y9WWTi5sffo&ab_channel=DrD%E2%80%99sMathHelp Study material- https://drive.google.com/drive/folders/1J_eV95KDwMDlGvvlGkCpEZM2Taial-Db References