Saba Bahouth 1 Supplement 6 Linear Programming

Saba Bahouth 2  Scheduling school busses to minimize total distance traveled when carrying students  Allocating police patrol units to high crime areas in order to minimize response time to 911 calls  Scheduling tellers at banks so that needs are met during each hour of the day while minimizing the total cost of labor  Picking blends of raw materials in feed mills to produce finished feed combinations at minimum costs  Selecting the product mix in a factory to make best use of machine and labor-hours available while maximizing the firm’s profit  Allocating space for a tenant mix in a new shopping mall so as to maximize revenues to the leasing company Examples of Successful LP Applications

Saba Bahouth 3 Simple Example and Solution We make 2 products: Panels and Doors Panel: Labor: 2 hrs/unit Material:3 #/unit Door: Labor: 4 hrs/unit Material:1 #/unit Available Resources: Labor:80 hrs Material:60 # Profit: $10 per Panel $ 8 per Door

Saba Bahouth 4 Enumeration for Simple Example

Saba Bahouth 5 Material - wood Labor - hrs X2 - Doors X1 - Panels Add Paint Constraint (Resource)

Saba Bahouth 6 Let # of Colonial lots be Let # of Western lots be 1)Wood: 2)Pressing Time: 3)Finishing Time: 4)Budget: Max. profit Example Solution Using Simplex

Saba Bahouth Optimal Solution: X 1 = X 2 = Profit = $ 12,945.20

Saba Bahouth 8

9 Requirements of a Linear Programming Problem  Must seek to maximize or minimize some quantity (the objective function)  Objectives and constraints must be expressible as linear equations or inequalities  Presence of restrictions or constraints - limits ability to achieve objective  Must be willing to accept divisibility  Must have a convex feasible space

Saba Bahouth 10

Saba Bahouth 11 You’re an analyst for a division of Kodak, which makes BW & color chemicals. At least 30 tons of BW and at least 20 tons of color must be made each month. The total chemicals made must be at least 60 tons. How many tons of each chemical should be made to minimize costs? Color: $ 3,000 manufacturing cost per ton per month BW: $2,500 manufacturing cost per ton per month Minimization Example

Saba Bahouth 12 Graphical Solution Tons, Color Chemical (X 2 ) Tons, BW Chemical (X 1 ) BW Color Total X1X1 X2X2 Find values for X 1 + X 2  60 X 1  30 X 2  20