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 4

Enumeration for Simple Example

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 8

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 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 BW: $2,500 manufacturing cost manufacturing cost per ton per monthper ton per month

Minimization Example

Saba Bahouth 12

Graphical Solution

0

20

40

60

80

0

Tons

, Col

or C

hem

ical (

X 2)

20 40 60 80Tons, BW Chemical (X1)

BW

Color

Total

X1

X2Find values for

X1 + X2 60

X1 30

X2 20