ECON 3300

1
ECON 3300 5

• 09/18/06

2
Example

• The Xecko tool company is considering a job for
  two airplane wing parts. Each wing part must be
  processed through three manufacturing stages
  stamping, drilling and finishing- for which
  company has limited available hours. The linear
  programming model to determine the number of part
  1 (x1) and part 2 (x2) the company should produce
  in order to maximize its profit as follows

3
Example

• Maximize Z650x1910x2
• Subject to
• 4x17.5x2lt105 (stamping hr)
• 6.2x14.9x2lt90 (drilling hr)
• 9.1x14.1x2lt110 (finishing hr)
• x1, x2gt0

4
Product Mix Example

• Quick Screen is a clothing manufacturing company
  that specializes in producing commemorative
  shirts immediately following major sporting
  events like the World Series, Super Bowl, and
  Final Four. The company has been contracted to
  produce a standard set of shirts for the winning
  team, either State University or Tech, following
  a college football game on New Years Day. The
  items produced include both sweatshirts, one with
  silk screen printing on the front and one with
  print on both sides, and two T-shirts of the same
  configuration. The company has to complete all
  production within 72 hours after the game, at
  which time a trailer truck will pick up the
  shirts. The company will work around the clock.
  The truck has enough capacity to accommodate
  1,200 standard-size boxes. A standard-size box
  holds 12

5
Product Mix Example

• T-shirts, whereas a box of one dozen sweatshirts
  is three times the size of a standard box. The
  company has budgeted 25,000 for the production
  run. It has 500 dozen blank sweatshirts and
  T-shirts each in stock ready for the production.

6
Product Mix Example
7
Product Mix Example

• Decision variables (number of dozen boxes)
• x1sweatshirts, front printing
• x2sweatshirts, back and front printing
• x3T-shirts, front printing
• x4T-shirts, back and front printing
• Constraints
• .10x1.25x1.08x3.21x4lt72hr
• 3x13x2x3x4lt1200 boxes
• 36x148x225x335x4lt25000
• x1x2lt500 dozen sweat shirts
• x3x4lt500 dozen T-shirts

8
Product Mix Example

• Objective function
• Maximize Z90x1125x245x365x4

9
Diet Example

• Breathtakers, a health and fitness center,
  operates a morning fitness program for senior
  citizens. The program includes aerobic exercises,
  either swimming or step exercise, followed by a
  healthy breakfast in the dinning room.
  Breathtakers dietitian wants to develop a
  breakfast that will be high in calories, calcium,
  protein, and fiber, which are especially
  important to senior citizens, but low in fat and
  cholesterol. She also wants to minimize cost. She
  has selected the following possible food items,
  whose individual nutrient contributions and cost
  from which to develop a standard breakfast menu
  are shown in the following table.

10
(No Transcript)
11
Diet Example

• The dietitian wants the breakfast to include at
  least 420 calories, 5 milligrams of iron, 400
  milligrams of calcium, 20 grams of protein, and
  12 grams of fiber. Furthermore she wants to limit
  fat to no more than 20 grams and cholesterol to
  30 milligrams.

12
Diet Example

• Decision variables
• x1cups of bran cereal
• x2cups of dry cereal
• x3cups of oatmeal
• x4cups of oat bran
• x5eggs
• x6slices of bacon
• x7oranges
• x8cups of milk
• x9cups of orange juice
• x10slices of wheat toast

13
Diet Example

• Constraints
• 90x1110x2100x390x475x535x665x7100x8120x96
  5x10gt420 (calories)
• 2x22x32x45x53x64x8x10lt20 (fat)
• 270x58x612x8lt30 (cholesterol)
• 6x14x22x33x4x5x7x10gt5 (iron)
• 20x148x212x38x430x552x7250x83x926x10gt40
  (calcium)
• 3x14x25x36x47x52x6x79x8x93x10gt20
  (protein)
• 5x12x23x34x4x73x10gt12 (fiber)

14
Diet Example

• Objective function
• minimize Z0.18x10.22x20.10x30.12x40.10x50.09
  x60.40x70.16x80.50x90.07x10

15
Marketing Example

• The Biggs Department Store chain has hired an
  advertising firm to determine the types and
  amount of advertising it should invest in for its
  stores. The three types of advertising available
  are television and radio commercials and
  newspaper ads. The retail chain desires to know
  the number of each type of advertisement it
  should purchase in order to maximize the
  exposure. It is estimated that each ad or
  commercial will reach the following potential
  audience and cost the following amount

16
Marketing Example
17
Marketing Example

• Company must consider the following resource
  constraints
• The budget limit for advertising is 100000.
• The television station has time available for 4
  commercials.
• The radio station has time available for 10
  commercials.
• The newspaper has space available for 7 ads.
• The advertising agency has time and staff
  available for producing no more than a total of
  15 commercials and /or ads.

18
Marketing Example

• Decision variables
• x1number of television commercials
• x2number of radio commercials
• x3number of newspaper ads
• Constraints
• 15000x16000x24000x3lt100000 (budget)
• x1lt4 (television commercials)
• x2lt10 (radio commercials)
• x3lt7 (newspaper ads)
• x1x2x3lt15 (commercial ads)

19
Marketing Example

• Objective function
• maximize Z20000x112000x29000x3

20
Transportation Example

• The Zephyr Television Company ships television
  from three warehouses to three retail stores on a
  monthly basis. Each warehouse has a fixed supply
  per month and each store has a fixed demand per
  month. The manufacturer wants to know the number
  of television sets to ship from each warehouse to
  each store in order to minimize the total cost of
  transportation.

21
Transportation Example

• Each warehouse has the following supply of
  televisions available for shipment each month
• Each retail store has the following monthly
  demand for television sets

22
Transportation Example

• Costs of transporting television sets from the
  warehouses to retail stores vary as a result of
  differences in modes of transportation and
  distances. The shipping cost per television set
  for each route is as follows

23
Transportation Example

• Decision variables
• xij number of television sets shipped from
  warehouse i to retail store j
• where i1, 2, 3 and jA, B, C
• Constraints
• x1Ax1Bx1Clt300
• x2Ax2Bx2Clt200
• x3Ax3Bx3Clt200
• x1Ax2Ax3A150
• x1Bx2Bx3B250
• x1Cx2Cx3C200

24
Transportation Example

• Objective function
• Minimize Z16x1A18x1B11x1C14x2A12x2B13x2C13x
  3A15x3B17x3C

25
Transportation Example

• Objective function
• Minimize Z16x1A18x1B11x1C14x2A12x2B13x2C13x
  3A15x3B17x3C