1 / 42

MBA, Semester 2 Operations Management Ms. Aarti Mehta Sharma

MBA, Semester 2 Operations Management Ms. Aarti Mehta Sharma. Linear programming. An efficient method for determining an optimal decision from a large no. of decisions. Selection of product mix Maximize profits, subject to constraints. Objective fn Z = c 1 x 1 + c 2 x 2 +…… c n x n

jane-mcdowell
Download Presentation

MBA, Semester 2 Operations Management Ms. Aarti Mehta Sharma

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MBA, Semester 2Operations ManagementMs. Aarti Mehta Sharma

  2. Linear programming

  3. An efficient method for determining an optimal decision from a large no. of decisions. Selection of product mix Maximize profits, subject to constraints

  4. Objective fn Z = c1x1 + c2x2 +…… cnxn c1,c2 are uncontrollable variables X1, x2 are different attributes

  5. Optimize (maximize or minimize) Z = c1x1 + c2x2 +…… cnxn Subject to linear constraints A11x1 + a12x2 + ….. + a1nxn (≤,=,>=) A21x1 + a22x2 + ….. + a2nxn (≤,=,>=) . Am1x1 + am2x2 + ….. + amnxn (≤,=,>=) X1,x2….xn ≥ 0 General modelof LPP

  6. Graphical method Simplex method

  7. A small scale industry manufactures electrical regulators, the assembly of which is being accomplished by a small group of skilled workers, both men and women. Due to the limitations of space and finance, the no. of workers employed cannot exceed 11 and their salary bill not exceed more than Rs 60,000 per month. Graphical method

  8. The male members of the skilled workers are paid Rs 6000 p.m. and female workers Rs 5000 p.m. data collected on the performance of these workers indicate that a male member contributes Rs 10,000 pm to total return of the industry and a female worker contributes Rs 8,500 p.m. determine the no. of male and female workers to be employed to maximize the monthly total return.

  9. X1 = male workers X2 = female workers Maximize Z = 10,000x1 + 8500 x2 X1 + x2 ≤ 11 6000 x1 + 5000 x2 ≤ 60,000 6x1 + 5 x2 ≤ 60 x1≥ 0; x2 ≥ 0

  10. Turn inequalities into equalities X1 + x2 ≤ 11 X1 + x2 = 11 when x1 =0, x2 = 11 when x1 =11, x2 = 0 6x1 + 5 x2 ≤ 60 6x1 + 5 x2 = 60 when x1 =0, x2 = 12 when x1 =10, x2 = 0

  11. O- 0,0 B – 10,0 P – 5,6 C – 0,11

  12. Maximize Z = 10,000x1 + 8500 x2 O ; 10,000 * 0 + 8500 * 0 = 0 B ; 10,000 * 10 + 8500 * 0 = 1,00,000 C ; 10,000 * 0 + 8500 * 11 = 93,500 O ; 10,000 * 5 + 8500 * 6 = 1,01,000 X1 = 5 X2 = 6

  13. A company manufactures two types of products P1 and P2. each product uses lathe and milling machines. The processing time per unit of P1 on the lathe is 5 hours and on the milling machine is 4 hours. The processing time per unit of P2 on the lathe is 10 hours and on the milling machine is 4 hours. The maximum no. of hours available per week on the lathe and the milling machine are 60 hrs and 40 hrs. The profit per unit of P1 = Rs 6.00 and P2 = Rs 8.00. Formulate a LP model to determine the production volume of Q

  14. Each of the products such that the total profit is maximised. Soln : maximize Z = 6x1 + 8 X2 subject to 5 x1 + 10 x2≤ 60 4 x1 + 4 x2≤ 40 x1 ≥ 0 ; x2 ≥0

  15. maximize Z = 6x1 + 8 X2 + 0s1 +0s2 subject to 5 x1 + 10 x2 = 60 4 x1 + 4 x2 = 40 x1 ≥ 0 ; x2 ≥0;

  16. A company manufactures two types of products P1 and P2. each product uses lathe and milling machines. The processing time per unit of P1 on the lathe is 5 hours and on the milling machine is 4 hours. The processing time per unit of P2 on the lathe is 10 hours and on the milling machine is 4 hours. The maximum no. of hours available per week on the lathe and the milling machine are 60 hrs and 40 hrs. The profit per unit of P1 = Rs 6.00 and P2 = Rs 8.00. Formulate a LP model to determine the production volume of Q

  17. Each of the products such that the total profit is maximised. Soln : maximize Z = 6x1 + 8 X2 subject to 5 x1 + 10 x2≤ 60 4 x1 + 4 x2≤ 40 x1 ≥ 0 ; x2 ≥0

  18. maximize Z = 6x1 + 8 X2 + 0s1 +0s2 subject to 5 x1 + 10 x2 + s1 = 60 4 x1 + 4 x2 + s2 = 40 x1 ≥ 0 ; x2 ≥0; s1,s2 ≥ 0 S1, s2 are called slack variables

  19. Continue till last row entries are all zero or negative Choose largest value in last row = 8 Column with largest value is known as pivotal column Divide values of soln column with values of pivotal colum Choose min – pivotal row 10 becomes pivot no. Reduce 10 to 1; reduce other values in col to 0

  20. Stop when all values in last row are 0 or negative The corresponding optimal solution is : X1(production volume of P1) = 8 units X2(production volume of P2) = 2 units

  21. Maximize Z= 10x1 + 15 x2 + 20 x 3 Subject to 2x1 +4x2 + 6x 3<= 24 3x1 + 9x2 + 6x3<= 30 x1,x2, x3 >=0 Case

  22. Maximize Z= 10x1 + 15 x2 + 20 x3 +0s1 + 0s2 Subject to 2x1 +4x2 + 6x 3 + s1 = 24 3x1 + 9x2 + 6x3 + s2 = 30 x1,x2, x3 s1, s2,s3 >=0 Where s1 & s2are slack variables Solution

  23. Since all the values of Cj-Zj are less than or equal to zero, the optimality is there and X1 = 6, X2 = 0; X3 = 2 and Zoptimum = 100

  24. A company makes two kinds of leather bags bag A and bag B. Bag A is a high quality bag and bag B is of lower quality. The respective profits are Rs. 4 and Rs. 3 per bag. The production of each of type A requires twice as much time as a bag of type B. if all bags were of type B, the company could make 1000 bags per day. Case

  25. The supply of leather is sufficient for only 800 bags per day (Both A & B combined). Bag A requires a fancy buckle and only 400 of these are available . There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt ? Formulate this problem as an LP problem and solve it using the simplex method.

  26. Maximize Z = 4x1 +3x2 Subject to the constraints 2x1 + x2 <=1000 X1 +x2 <= 800 X1<=400 X2<= 700 and x1,x2 >=0

  27. Maximize Z = 4x1 +3x2 +0s1 + 0s2 +0s3 +0s4 Subject to the constraints 2x1 + x2 + s1 =1000 X1 +x2 + s2= 800 X1+ s3 =400 X2 + s4 = 700 and x1,x2,s1,s2,s3,s4 >=0

  28. R2(new) R2(old)-R1(new) R4(new)R4(old)-R1(new)

  29. R1(new) R1(old) + 2R2(new) R3(new) R3(old) – R2(new) R4(new) R4(old) – 2R2

  30. Since Cj – Zj <= 0, the current solution cannot be improved upon. Thus , the company must manufacture x1 = 200 bags of type A and x2 = 600 bags of type B to obtain the max profit of Rs. 2,600.

More Related