1 / 24

INTRODUCTION TO LINEAR PROGRAMMING

INTRODUCTION TO LINEAR PROGRAMMING. CONTENTS Introduction to Linear Programming Applications of Linear Programming Reference: Chapter 1 in BJS book. A Typical Linear Programming Problem. Linear Programming Formulation: Minimize c 1 x 1 + c 2 x 2 + c 3 x 3 + …. + c n x n subject to

jersey
Download Presentation

INTRODUCTION TO LINEAR PROGRAMMING

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. INTRODUCTION TO LINEAR PROGRAMMING CONTENTS • Introduction to Linear Programming • Applications of Linear Programming Reference: Chapter 1 in BJS book.

  2. A Typical Linear Programming Problem Linear Programming Formulation: Minimize c1x1 + c2x2 + c3x3 + …. + cnxn subject to a11x1 + a12x2 + a13x3 + …. + a1nxn  b1 a21x1 + a22x2 + a23x3 + …. + a2nxn  b2 : : am1x1 + am2x2 + am3x3 + …. + amnxn bm x1, x2, x3 , …., xn 0 or, Minimize j=1, n cjxj subject to j=1, n aijxj -xn+i = bi for all i = 1, …, m xj  0 for all j =1, …, n

  3. Matrix Notation Minimize cx subject to Ax = b x  0 where

  4. An Example of a LP • Giapetto’s woodcarving manufactures two types of wooden toys: soldiers and trains Constraints: • 100 finishing hour per week available • 80 carpentry hours per week available • produce no more than 40 soldiers per week • Objective: maximize profit

  5. An Example of a LP (cont.) Linear Programming formulation: Maximize z = 3x1 + 2x2(Obj. Func.) subject to 2x1 + x2 100 (Finishing constraint) x1 + x2 80 (Carpentry constraint) x1 40 (Bound on soldiers) x1 0 (Sign restriction) x2 0 (Sign restriction)

  6. Assumptions of Linear Programming • Proportionality Assumption • Contribution of a variable is proportional to its value. • Additivity Assumptions • Contributions of variables are independent. • Divisibility Assumption • Decision variables can take fractional values. • Certainty Assumption • Each parameter is known with certainty.

  7. Linear Programming Modeling and Examples • Stages of an application: • Problem formulation • Mathematical model • Deriving a solution • Model testing and analysis • Implementation

  8. Capital Budgeting Problem • Five different investment opportunities are available for investment. • Fraction of investments can be bought. • Money available for investment: Time 0: $40 million Time 1: $20 million • Maximize the NPV of all investments.

  9. Transportation Problem • The Brazilian coffee company processes coffee beans into coffee at m plants. The production capacity at plant i is ai. • The coffee is shipped every week to n warehouses in major cities for retail, distribution, and exporting. The demand at warehouse j is bj. • The unit shipping cost from plant i to warehouse j is cij. • It is desired to find the production-shipping pattern xij from plant i to warehouse j, i = 1, .. , m, j = 1, …, n, that minimizes the overall shipping cost.

  10. Static Workforce Scheduling • Number of full time employees on different days of the week are given below. • Each employee must work five consecutive days and then receive two days off. • The schedule must meet the requirements by minimizing the total number of full time employees.

  11. Multi-Period Financial Models • Determine investment strategy for the next three years • Money available for investment at time 0 = $100,000 • Investments available : A, B, C, D & E • No more than $75,000 in one invest • Uninvested cash earns 8% interest • Cash flow of these investments:

  12. Cutting Stock Problem • A manufacturer of metal sheets produces rolls of standard fixed width w and of standard length l. • A large order is placed by a customer who needs sheets of width w and varying lengths. He needs bi sheets of length li, i = 1, …, m. • The manufacturer would like to cut standard rolls in such a way as to satisfy the order and to minimize the waste. • Since scrap pieces are useless to the manufacturer, the objective is to minimize the number of rolls needed to satisfy the order.

  13. Multi-Period Workforce Scheduling • Requirement of skilled repair time (in hours) is given below. • At the beginning of the period, 50 skilled technicians are available. • Each technician is paid $2,000 and works up to 160 hrs per month. • Each month 5% of the technicians leave. • A new technician needs one month of training, is paid $1,000 per month, and requires 50 hours of supervision of a trained technician. • Meet the service requirement at minimum cost.

  14. Solution: Capital Budgeting Problem Decision Variables: xi: fraction of investment i purchased Formulation: Maximize z = 13x1 + 16x2 + 16x3 + 14x4 + 39x5 subject to 11x1 + 53x2 + 5x3 + 5x4 + 29x5 £ 40 3x1 + 6x2 + 5x3 + x4 + 34x5 £ 20 x1£ 1 x2£ 1 x3£ 1 x4£ 1 x5£ 1 x1, x2, x3, x4, x5³ 0

  15. Solution: Transportation Problem Decision Variables: xij: amount shipped from plant i to warehouse j Formulation: Minimize z = subject to = ai, i = 1, … , m  bj, j = 1, … , n xij 0, i = 1, … , m, j = 1, … , n

  16. Solution: Static Workforce Scheduling LP Formulation: Min. z = x1+ x2 + x3 + x4 + x5 + x6 + x7 subject to x1 + x4 + x5 + x6 + x7³17 x1+ x2 + x5 + x6 + x7³13 x1+ x2 + x3 + x6 + x7 ³15 x1+ x2 + x3 + x4 + x7³19 x1+ x2 + x3 + x4 + x5³14 x2 + x3 + x4 + x5 + x6³16 x3 + x4 + x5 + x6 + x7³11 x1, x2, x3, x4, x5, x6, x7³ 0

  17. Solution: Multiperiod Financial Model Decision Variables: A, B, C, D, E : Dollars invested in the investments A, B, C, D, and E St: Dollars invested in money market fund at time t (t = 0, 1, 2) Formulation: Maximize z = B+ 1.9D+ 1.5E+ 1.08S2 subject to A+ C+ D+ S0 = 100,000 0.5A+ 1.2C+ 1.08S0 = B + S1 A+ 0.5B+ 1.08S1 = E + S2 A £ 75,000 B £ 75,000 C £ 75,000 D £ 75,000 E £ 75,000 A, B, C, D, E, S0, S1, S2³ 0

  18. Solution: Multiperiod Workforce Scheduling Decision Variables: xt: number of technicians trained in period t yt: number of experienced technicians in period t Formulation: Minimize z = 1000(x1 + x2 + x3 + x4 + x5) + 2000(y1 + y2 + y3 + y4 + y5) subject to 160y1 - 50 x1³ 6000 y1 = 50 160y2 - 50 x2³ 7000 0.95y1 + x1 = y2 160y3 - 50 x3³ 8000 0.95y2 + x2 = y3 160y4 - 50 x4³ 9500 0.95y3 + x3 = y4 160y5 - 50 x5³11000 0.95y4 + x4 = y5 xt, yt³ 0, t = 1, 2, 3, … , 5

  19. Cutting Stock Problem (contd.) • Given a standard sheet of length l, there are many ways of cutting it. Each such way is called a cutting pattern. • The jth cutting pattern is characterized by the column vector aj, where the ith component, namely, aij, is a nonnegative integer denoting the number of sheets of length li in the jth pattern. • Note that the vector aj represents a cutting pattern if and only if i=1,n aijli  l and each aij is a nonnegative number.

  20. Cutting Stock Problem (contd.) Formulation: Minimize i=1,n xi subject to i=1,n aij xi  bi i = 1, …, m xi  0 j = 1, …, n xi integer j = 1, …, n

  21. Feasible Region Feasible Region: Set of all points satisfying all the constraints and all the sign restrictions Example: Max. z = 3x1 + 2x2 subject to 2x1 + x2£ 100 x1 + x2£ 80 x1 £ 40 x1 ³ 0 x2 ³ 0

  22. Example 1 Maximize z = 50x1 + 100x2 subject to 7x1 + 2x2³ 28 2x1 + 12x2³ 24 x1, x2 ³ 0 Feasible region in this example is unbounded.

  23. Example 2 Maximize z = 3x1+ 2x2 subject to 1/40x1 + 1/60x2£ 1 1/50x1 + 1/50x2£ 1 x1 ³ 30 x2 ³ 20 x1, x2 ³ 0 This linear program does not have any feasible solutions.

  24. Example 3 Max. z = 3x1 + 2x2 subject to 1/40 x1 + 1/60x2£ 1 1/50 x1 + 1/50x2£ 1 x1, x2 ³ 0 This linear program has multiple or alternative optimal solutions.

More Related