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2TN – Linear Programming

2TN – Linear Programming. Linear Programming. Linear Programming Discussion. Requirements of a Linear Programming Problem Formulate: Determine:Graphical Solution to a Linear Programming Problem. What is Linear Programming?. Mathematical technique Not computer programming

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2TN – Linear Programming

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  1. 2TN – Linear Programming • Linear Programming

  2. Linear Programming Discussion • Requirements of a Linear Programming Problem • Formulate: • Determine:Graphical Solution to a Linear Programming Problem

  3. What is Linear Programming? • Mathematical technique • Not computer programming • Allocates scarce resources to achieve an objective • Pioneered by George Dantzig in World War II

  4. Linear ProgrammingGeneral Discussion • Resources are constrained or limited. • Model has an objective (function) • subject to constraints. • Linearity

  5. Linear Programming Applications • Given machine and labor hours • Given demand • Given limited patrol cars • Given minimum daily diet requirements

  6. Requirements of a Linear Programming Problem • Must seek to maximize or minimize some quantity • Presence of restrictions or constraints – • Must be alternative courses of action to choose from • Objectives and constraints must be expressible as linear equations or inequalities

  7. Objective Function • Cj is a constant that describes the rate of contribution to costs or profit of units being produced (Xj). • Z is the total cost or profit from the given number of units being produced. Maximize (or Minimize) Z = C1X1 + C2X2 + ... + CnXn

  8. Constraints A11X1 + A12X2 + ... + A1nXnB1 A21X1 + A22X2 + ... + A2nXn B2 : : AM1X1 + AM2X2 + ... + AMnXn=BM • Aij are resource requirements for each of the related (Xj) decision variables. • Bi are the available resource requirements. • Note that the direction of the inequalities can be all or a combination of , , or = linear mathematical expressions.

  9. Non-Negativity Requirement • All linear programming model formulations require their decision variables to be non-negative. • While these non-negativity requirements take the form of a constraint, they are considered a mathematical requirement to complete the formulation of an LP model. X1,X2, …, Xn 0

  10. Graphical Solution Method2 Variables • Step 1 - Draw graph with vertical & horizontal axes • (1st quadrant only) • Step 2 - Plot constraints as lines • Use (X1,0), (0,X2) for line • Step 3 - Plot constraints as planes • Use < or > signs • Step 4 - Find feasible region • Step 5 - Find optimal solution • Objective function plotted • Step 6 – Calculate optimized value

  11. ELECTRONIC COMPANY PROBLEM Hours Required to Produce 1 Unit X1 X2 Available Hours Departments Walkmans Watch-TV’s This Week Electronics 4 3 240 Assembly 2 1 100 Profit/unit $7 $5 Constraints: 4x1 + 3x2 £ 240 (Hours of Electronic Time) 2x1 + 1x2£ 100 (Hours of Assembly Time) Objective: Maximize: 7x1 + 5x2

  12. 120 100 80 60 40 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X1) Step 1 – Draw Graph Number of Watch-TVs (X2)

  13. Step 5 - Find optimal solution • Plot function line

  14. Step 5 - Find optimal solution (Cont’d)In This Case:Calculate the point where both constraint lines intersect

  15. Step 5 - Find optimal solution (Cont’d)

  16. Step 5 - Find optimal solution (Cont’d) Electronics Department 120 100 Assembly Department 80 Number of Watch-TVs (X2) 60 40 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X1)

  17. Step 6 – Calculate optimized value Plug in values for X1 and X2 Therefore: the best profit scenario is $410.00

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