Introduction to OR

1
Introduction to OR2

• Industrial Engineering
• Prof. Yang

2
Linear Programming

• '47 G.B. Dantzig Simplex Method
• '79 L.G. Katchian Ellipsoid Method
• '84 Karmarkar Projection Method

3

• Variable A factor subject to change within
  problem
• Decision (Control) Var. under the control of DM
  and could have an impact on the solution to the
  problem of interest.
• Objective desires of DM - max, min
• Goal combination of an obj. with a target
• Constraint absolute or inflexible goal

4
A House Insulation Problem

• 1. Amount of attic insulation installed.
• 2. Amt of side-wall insul. to be installed.
• 3. Amt of caulking to be done.
• 4. No. of storm windows.
• 5. No. of insulating drapes used.
• 6. Amt of insul. around hot-water tank.
• 7. Temperatures experienced.
• 8. Wind velocity and direction.
• 9. Amt of sunshine incident on house.
• 10. No of individuals within the house.
• 11. Frequency of door openings
• 12. Cost of utilities furnished.

5
EX 2-1 Sam Young Ind. CO.

• SY is producing knives and forks for dining
  table.
• Objective Maximize Total Weekly Profit
• Due to the excellent quality, all amount produced
  are exported overseas.
• Profits per case
• Knife \8,000 / case
• Fork \6,000 / case
• Knives and forks need to be process on PRESS and
  POLISH.
• Availabilities of PRESS and POLISH times are
  estimated to be 700 hours and 1000 hours per
  week, respectively.
• Both products would be competing for the same
  production capacity. It requires 12 min of PRESS
  and 30 min of POLISH to produce a case of knives
  and 24 min of PRESS and 15 min of POLISH to
  produce a case of forks.

6

• The Mkting Dept has concluded that the company
  could sell as much of either product as could be
  produced with the available capacity. However,
  because both products would be competing for the
  same production capacity in PRESS and POLISH, it
  is not clear which mix between the two products
  would be most profitable.
• Capacity Used Per Unit Production Rate

Process
Knife
Fork
Available
PRESS
12 min
24 min
700 hr
POLISH
30 min
15 min
1000 hr
\ 8,000
\ 6,000
Unit Profit
x1
x2
7
LP Standard Form

• LP Standard Form
•  
• Max Z c1x1 c2x2 cnxn
•  
• subject to
•   a11x1 a12x2 c1nxn b1
• a21x1 a22x2 c2nxn b2
• am1x1 am2x2 cmnxn bm
•   x1 0, x2 0, , xn 0

8
EX 2 RADIATION THERAPY (p 43)

• Fract. Of Entry
• Dose Absorbed by Area
• Area Beam 1 Beam2 Restr on Total Avg. Dosage
• Healthy anatomy 0.4 0.5 Minimize
• Critical tissues 0.3 0.1 2.7
• Tumor 0.5 0.5 6
• Center of Tumor 0.6 0.4 6

9
Ex3 REGIONAL PLANNING

• Kibbutzim, or communal farming communities, in
  Israel. SOUTHERN CONFEDERATION OF KIBBUTZIM
  (SCK). The planning office is planning
  agricultural production for the coming year. The
  agricultural output of each kibbutz is limited by
  both the amount of available irrigable land and
  by the quantity of water allocated for irrigation
  by the Water Commissioner. These data are given
  in Table 1. The crops being considered for the
  upcoming season are sugar beets, cotton, and
  sorghum. These crops differ primarily in their
  expected net return per acre and their
  consumption of water. In addition, the Ministry
  of Ag. has set a max quota for the total acreage
  that can be devoted to each of these crops by the
  SCK as shown in Table 2. The three kibbutzim
  belonging to the SC have agreed that every
  kibbutz will plant the same proportion of its
  available irrigable land. For example, if
  kibbutz 1 plants 200 of its available 400 acres,
  then kibbutz 2 must plant 300 of its 600 acres,
  while kibbutz 3 plants 150 acres of its 300
  acres. However, any combination of the crops may
  be grown at any of the kibbutzim. The job facing
  the Coordinating Office is to plan how many acres
  to devote to each crop at the respective
  kibbutzim while satisfying the given
  restrictions. The objective is to maximize the
  total net return to the Southern Confederation as
  a whole.

10
Tables

• Table 1 Resource Data

Usable Land (Acre)
Water Alloc. (Acre Feet)
Kibbutz
1
400
600
2
600
800
3
300
375
Table 2 Crop Data
Max Qta (Ac)
Water Alloc. (Acre Feet)
Max Qta (Ac)
Crop
Sg Beets
00
3
400
Cotton
500
2
300
Sorghum
325
1
100
11
EXAMPLE 4

• Suppose that a custom molder has one
  injection-molding machine and two different dies
  to fit the machine. Due to differences in number
  of cavities and cycle times, with the first die
  he can produce 100 cases of six-ounce juice
  glasses in six hours, while with the second die
  he can produce 100 cases of ten-ounce fancy
  cocktail glasses in five hours. He prefers to
  operate only on a schedule of 60 hours of
  production per week. He stores the week's
  production in his own stockroom where he has an
  effective capacity of 15,000 cubit feet. A case
  of six-ounce juice glasses requires 10 cubit feet
  of storage space, while a case of ten-ounce juice
  glasses requires 20 cubic feet due to special
  packaging. The contribution of the six-ounce
  juice glasses is 5.00 per case however, the
  only customer available will not accept more than
  800 cases per week. The contribution of the
  ten-ounce cocktail glasses is 4.50 per case and
  there is no limit on the amount that can be sold.
  How many cases of each type of glass should be
  produced each week in order to maximize the total
  contribution?