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 DoorsPanel: Labor: 2 hrs/unit
Material: 3 #/unitDoor: Labor: 4 hrs/unit
Material: 1 #/unit
Available Resources: Labor: 80 hrsMaterial: 60 #
Profit: $10 per Panel$ 8 per Door
Saba Bahouth 5
Material - wood
Labor - hrs
X2 - Doors
X1 - Panels0
20
60
20 40
40
28
31.43
228
10
QuartsXX 2246.58 21
QuartsXX 1766.58 21
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
2X
000,55020 21 XX
1X
40023 21 XX
50043 21 XX
000,775.4350 21 XX
21 10080 XXZ
Example Solution Using Simplex
Saba Bahouth 70 50 100 150 200 250 1X
2X
250
200
150
100
50
40023 21 XX
8000Z
700075.4350 21 XX
50043 21 XX
21 10080)( XXZMax
50005020 21 XX
Optimal Solution:X1 = 89.09X2 = 58.18Profit = $ 12,945.20
Saba Bahouth 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 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 BW: $2,500 manufacturing cost manufacturing cost per ton per monthper ton per month
Minimization Example