1. 1 2TN – Linear Programming  Linear Programming

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

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

4. 4 Linear Programming General Discussion  Resources are constrained or limited.  Model has an objective (function) –subject to constraints.  Linearity

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

6. 6 Requirements of a Linear Programming Problem 1 Must seek to maximize or minimize some quantity 1 Presence of restrictions or constraints – 1 Must be alternative courses of action to choose from 4 Objectives and constraints must be expressible as linear equations or inequalities

7. 7 Objective Function Maximize (or Minimize) Z = C 1 X 1 + C 2 X 2 +... + C n X n  C j is a constant that describes the rate of contribution to costs or profit of units being produced (X j ).  Z is the total cost or profit from the given number of units being produced.

8. 8 Constraints A 11 X 1 + A 12 X 2 +... + A 1n X n  B 1 A 21 X 1 + A 22 X 2 +... + A 2n X n  B 2 : A M1 X 1 + A M2 X 2 +... + A Mn X n = B M  A ij are resource requirements for each of the related (X j ) decision variables.  B i are the available resource requirements.  Note that the direction of the inequalities can be all or a combination of , , or = linear mathematical expressions.

9. 9 Non-Negativity Requirement X 1,X 2, …, X n  0  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.

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

11. 11 ELECTRONIC COMPANY PROBLEM Hours Required to Produce 1 Unit Departments X1X1 Walkmans X2X2 Watch-TV’s Available Hours This Week Electronics43240 Assembly21100 Profit/unit$7$5 Constraints: 4x 1 + 3x 2  240 (Hours of Electronic Time) 2x 1 + 1x 2  100 (Hours of Assembly Time) Objective:Maximize: 7x 1 + 5x 2

12. 12 Step 1 – Draw Graph Number of Watch-TVs (X 2 ) 0 20 40 60 80 100 120 01020304050607080 Number of Walkmans (X 1 )

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

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

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

16. 16 Step 5 - Find optimal solution (Cont’d) 0 20 40 60 80 100 120 01020304050607080 Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Electronics Department Assembly Department

17. 17 Step 6 – Calculate optimized value Therefore: the best profit scenario is $410.00 Plug in values for X 1 and X 2