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Supplement 6 Linear Programming

Supplement 6 Linear Programming. Examples of Successful LP Applications. 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

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Supplement 6 Linear Programming

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  1. Supplement 6Linear Programming

  2. Examples of Successful LP Applications • 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

  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

  4. Enumeration for Simple Example

  5. X2 - Doors 60 Add Paint Constraint (Resource) Material - wood 40 31.43 20 10 Labor - hrs X1 - Panels 0 8 20 22 28 40

  6. Example Solution Using Simplex • Let # of Colonial lots be • Let # of Western lots be • Wood: • Pressing Time: • Finishing Time: • Budget: • Max. profit

  7. 250 200 150 100 50 Optimal Solution: X1 = 89.09 X2 = 58.18 Profit = $ 12,945.20 0 50 100 150 200 250

  8. Requirements of a Linear Programming Problem • Must seek to maximize or minimizesome 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

  9. Minimization Example BW: $2,500 manufacturing cost per ton per month 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

  10. Graphical Solution Find values for X1 + X2 60 X1  30 X2  20 X2 BW 80 60 Total 40 Tons, Color Chemical (X2) 20 Color X1 0 0 20 40 60 80 Tons, BW Chemical (X1)

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