1 / 16

Operations Research I Chapter 02 Modeling with Linear Programming

Operations Research I Chapter 02 Modeling with Linear Programming. Dr. Ayham Jaaron First semester 2013/2014 August 2013 . Two-Variable LP Model. This section deals with formulation of a two-variable LP.

michi
Download Presentation

Operations Research I Chapter 02 Modeling with 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. Operations Research IChapter 02Modeling with Linear Programming Dr. AyhamJaaron First semester 2013/2014 August 2013

  2. Two-Variable LP Model • This section deals with formulation of a two-variable LP. • We will learn and practice the formulation process before we can try to solve real-life problems.

  3. LP Model Formulation: Two variable LP Model A Maximization Example (1 of 4) • Product mix problem - Beaver Creek Pottery Company • How many bowls and mugs should be produced to maximize profits given labor and materials constraints? • Product maximum daily resource requirements availability and unit profit is given by:

  4. LP Model Formulation A Maximization Example (2 of 4)

  5. LP Model Formulation A Maximization Example (3 of 4) Resource 40 hrs of labor per day Availability: 120 lbs of clay Decision x1 = number of bowls to produce per day Variables: x2 = number of mugs to produce per day Objective Maximize Z = $40x1 + $50x2 Function: Where Z = profit per day Resource 1x1 + 2x2 40 hours of labor Constraints: 4x1 + 3x2 120 pounds of clay Non-Negativity x1  0; x2  0 Constraints:

  6. LP Model Formulation A Maximization Example (4 of 4) Complete Linear Programming Model: Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2  40 4x 1+ 3x2  120 x1, x2  0 Can we solve it? What do you think? Shall we try?

  7. Feasible Solutions A feasible solution does not violate any of the constraints: Example: x1 = 5 bowls x2 = 10 mugs Z = $40x1 + $50x2 = $700 Labor constraint check: 1(5) + 2(10) = 25 < 40 hours Clay constraint check: 4(5) + 3(10) = 50<120 pounds

  8. Infeasible Solutions An infeasible solution violates at least one of the constraints: Example: x1 = 10 bowls x2 = 20 mugs Z = $40x1 + $50x2 = $1400 Labor constraint check: 1(10) + 2(20) = 50 > 40 hours

  9. Example 2: The Reddy Mikks Company • Construction of the LP model (from book page 12) Reddy Mikks produces both interior and exterior paints from two raw materials, M1&M2. The following table provides the basic data of the problem.

  10. Example 2 .. Cont’d (class task: 10 minutes) • A market survey indicates that the daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton. Also, the maximum daily demand of interior paint is 2 ton. • Reddy Mikks wants to determine the optimum (best) product mix of interior and exterior paints that maximizes the total daily profit. CLASS TASK: TRY

  11. Problem Formulation • Decision variables X1= Tons produced daily of exterior paint. X2= Tons produced daily of interior paint. • Objective Function Maximize Z= 5 X1 + 4 X2 • Constraints 6 X1+4 X2 24 X1+2 X2 6 - X1+ X2 1 X2 2 X1, X2 0

  12. TRY AT CALSSExample 3: Calculators Company (1 of 4) • A calculator company produces scientific calculators and graphing calculators. There is a demand forecast of at least 100 scientific calculators and 80 graphing calculators per day. • Manufacturing capacity says that a maximum of 200 scientific calculators and 170 graphing calculators can be produced each day. There is a contract that at least  200 calculators would be produced each day. • If each scientific calculator results in the loss of $2 and each graphing calculator results in a profit of $5, how many of each should be produced each day for the maximum profit?

  13. Problem Formulation (2 of 4) • Let x be the number of scientific calculators. • Let Y be the number of graphing calculators. • Each scientific calculator yields a loss of $2 = -2 • Each graphing calculator yields a profit of $5 = 5 • we have to maximize the profit : Maximize : z= -2x + 5y

  14. Problem Formulation..Con’t (3 of 4) • now, let us check the constraints : • minimum of 100 scientific calculators and 80 graphing calculators • maximum of 200 scientific calculators and 170 graphing calculators 100 ≤ x ≤ 200 80 ≤ y ≤ 170 • contract of at least 200 calculators per day x + y ≥ 200

  15. Complete LP Model..Con’t (4 of 4) • Decision variables X= no. of Scientific Calculators Y= no. of Graphical calculators • Objective Function Maximize Z= -2X + 5 Y • Constraints 100 ≤ x ≤ 200 80 ≤ y ≤ 170 x + y ≥ 200 X, Y ≥ 0

  16. More than just mathematics • In some cases, a "common sense" solution may be reached through simple observations. Examples below (see the book for details) • Slow elevator • Check-in facilities at a large airport (British culture) • In a steel mill (Employees delays)

More Related