Classic LP Examples

1
Classic LP Examples

• Topics
• Employee scheduling problem
• Energy distribution problem
• Feed mix problem
• Cutting stock problem
• Regression analysis
• Model Transformations

2
(More) Examples of LP Formulations
1. Employee Scheduling
Macrosoft has a 24-hour-a-day, 7-days-a-week toll
free hotline that is being set up to answer
questions regarding a new product. The following
table summarizes the number of full-time
equivalent employees (FTEs) that must be on duty
in each time block.
FTEs
Shift
Time
1
0-4
15
2
4-8
10
3
8-12
40
4
12-16
70
5
16-20
40
6
20-0
35
3
Constraints for Employee Scheduling

• Macrosoft may hire both full-time and part-time
  employees. The former work 8-hour shifts and the
  latter work 4-hour shifts their respective
  hourly wages are 15.20 and 12.95. Employees may
  start work only at the beginning of one of 6
  shifts.
• At least two-thirds of the employees working at
  any one time must be full-time employees.
• Part-time employees can only answer 5 calls in
  the time a full-time employee can answer 6 calls.
  (i.e., a part-time employee is only 5/6 of a
  full-time employee.)

Formulate an LP to determine how to staff the
hotline at minimum cost.
4
Decision Variables
xt of full-time employees that begin work in
shift t yt of part-time employees that work
shift t
(4 ? 12.95)
(8 ? 15.20)
min
121.6 (x1 x6)
51.8 (y1 y6)
5
³
s.t.
y1

15
x6 x1

6
5
³
y2

10
x1 x2

All shifts must be covered
6
5
³
y3

40
x2 x3

6
5
³
y4

70
x3 x4

6
5
³
y5

40
x4 x5

6
5
³
y6

35
x5 x6

6
PT employee is 5/6 FT employee
5
More constraints
At least 2/3 workers must be full time
2
(x6 x1)
³
(x6 x1 y1)

3
2
³
(x1 x2 y2)
(x1 x2)

3
.
.
.
2
³
(x5 x6)
(x5 x6 y6)
3
Nonnegativity
xt ³ 0, yt ³ 0 t 1,2, ,6
6
2. Energy Generation Problem (with piecewise
linear objective)

A Municipal Power and Light (MPL) would like to
determine optimal operating levels for their
electric generators and associated distribution
patterns that will satisfy customer demand.
Consider the following prototype system
Demand requirements
4 MW
1
1
Demand sectors
Plants
7 MW
2
2
3
6 MW
The two plants (generators) have the following
(nonlinear) efficiencies
Plant 1
0, 6 MW
6MW, 10MW
Unit cost (/MW)
10
25
Plant 2
0, 5 MW
5MW, 11MW
Unit cost (/MW)
8
28
The table is to be read as follows. For plant 1
if you generate at a rate of 8MW then the cost
(/sec) is 10(6) 25(2) 110.
7
Problem Statement and Notation
Formulate an LP that, when solved, will yield
optimal power generation and distribution levels.
Decision Variables
x
power generated at plant
1 at operating level 1
11
1
x
²
2
²
²
²
²
²
²
12

2
1
x
²
²
²
²
²
²
²
21


2
x
2
²
²
²
²
²
²
²

22

y
1 to demand sector 1
power sent from plant
11
2
y
1 ² ² ²


12
²
²
²
²
y
3
1

²
²
²

13
²
²
²
²

y
1
2
²
²
²

²
²
²
²
21

y
2
2
²
²
²

²
²
²
²
22

y
3
2
²
²
²

23
²
²
²
²

8
Formulation
8x
28x
Min
10x
25x





21
22
11
12
s.t.
y

y
x
x
y





11
12
13
11
12
y

x
x
y
y






21
22
23
21
22
y
y
4




11
21
y
y
7




12
22
y
y
6




13
23
6,



x
0

x

4


0



11
12


x
5,

x
0

6



0



22
21
y
y
y
y
y
y
³

0
,
,
,
,
,

11
12
13
21
22
23
Note that we can model the nonlinear operating
costs with an LP only because the efficiencies
have the right kind of structure. In particular,
the plant is less efficient (more costly) at
higher operating levels. Thus the LP solution
will automatically select level 1 first.
9
General Formulation of Power Distribution Problem
The above formulation can be generalized for any
number of plants, demand sectors, and generation
levels.
Indices/Sets
plants
i Î I
j Î J
demand sectors
generation levels
k Î K
Data
unit generation cost (/MW) for plant i at level k
Cik

upper bound (MW) for plant i at level k
Uik

dj

demand (MW) in sector j
Decision Variables
xik
power (MW) generated at plant i at level k
yij
power (MW) sent from plant i to sector j
10
General Network Formulation
å
å
cikxik
min


kÎK
iÎI
å
å
xik
s.t.
yij

" i Î I

j ÎJ
k ÎK
å
yij
dj
" j Î J

i ÎI

• xik uik " i Î I, k Î K
• 0 yij " i Î I, j Î J

11
3. Feed Mix Problem

• An agricultural mill produces three types of feed
  for cattle, sheep, and chickens by mixing the
  following raw ingredients corn, limestone,
  soybeans, and fish meal.
• These ingredients contain the following
  nutrients vitamins, protein, calcium, and crude
  fat in the following quantities

Nutrient
Vitamins
Protein
Calcium
Crude Fat
Ingredient
Corn
8
10
6
8
Limestone
6
5
10
6
Soybeans
10
12
6
6
Fish Meal
4
18
6
9

• Let aik quantity of nutrient k per kg of
  ingredient i

12
Constraints

• The mill has (firm) contracts for the following
  demands.

Demand (kg)
Cattle
Sheep
Chicken
dj
10,000
6,000
8,000

• There are limited availabilities of the raw
  ingredients.

Supply (kg)
Corn
Limestone
Soybeans
Fish Meal
si
6,000
10,000
4,0
00
5,000

• The different feeds have quality bounds per
  kilogram.

Vitamins
Crude fat
Protein
Calcium
min max
min max
min max
min max
Cattle
6 --
6 --
7 --
4 8
Sheep
6 --
6 --
6 --
4 8
6 --
Chicken
4 6
6 --
4 8

• The above values represent bounds Ljk Ujk

13
Costs and Notation

• Cost per kg of the raw ingredients is as follows

Soybeans
Fish meal
Limestone
Corn
cost/kg, ci
24
12
20
12
Formulate problem as a linear program whose
solution yields desired feed production levels at
minimum cost.
Indices/sets
i ? I
ingredients corn, limestone, soybeans, fish
meal
j ? J
products cattle, sheep, chicken feeds
k ? K
nutrients vitamins, protein, calcium, crude fat
14
Data
dj
demand for product j (kg)
si
supply of ingredient i (kg)
Ljk
lower bound on number of nutrients of type k per
kg of product j
upper bound on number of nutrients of type k per
kg of product j
Ujk
ci
cost per kg of ingredient i
aik
number of nutrients k per kg of ingredient i

Decision Variables
xij
amount (kg) of ingredient i used in producing
product j
15
LP Formulation
å
å
cixij
min




iÎI
jÎJ
å
xij
dj

s.t.

" j ? J


iÎI
xij
å

si
" i ? I

jÎJ
å
aikxij
" j ? J, k?K
Ujkdj

iÎI
å
aikxij
³
Ljk
dj
" j ? J, k?K


iÎI
" i ? I, j ? J
xij ³ 0
16
Generalization of feed Mix Problem Gives Blending
Problems
Blended commodities
Raw Materials
Qualities
corn, limestone,
protein, vitamins,
feed
soybeans, fish meal
calcium, crude fat
butane, catalytic
octane, volatility,
gasoline
reformate,
vapor pressure
heavy naphtha
pig iron,
carbon,
metals
ferro-silicon,
manganese,
carbide, various
chrome content
alloys
³
³
³
2 raw ingredients
1 quality
1 commodity
17
4. Trim-Loss or Cutting Stock problem

• Three special orders for rolls of paper have been
  placed at a paper mill. The orders are to be cut
  from standard rolls of 10 and 20 widths.

Length
Width
Order
1

5
10,000
2

7
30,000
3

9
20,000

• Assumption Lengthwise strips can be taped
  together
• Goal Throw away as little as possible

18
Problem What is trim-loss?

20

10
5000'
5'

5
9'

7

Decision variables xj length of roll cut using
pattern, j 1, 2, ?
19
Patterns possible


10
roll
20
roll
x1
x2
x3
x8
x9
x4
x5
x6
x7
5
2
0
0
4
2
2
1
0
0
7
0
1
0
0
1
0
2
1
0
9
0
0
1
0
0
1
0
1
2
Trim loss
0
3
1
0
3
1
1
4
2
min
z
10(x
x
x
) 20(x
x
x
x
x
x
)
1
2
7
9
3
4
5
6
8
³
s.t.
2x

4x

2x
2x
x

10,000




1
7
4
5
6
x
³
x

x

2x

30,000



7
8
2
5
³
x
x

x
2x

20,000



8
3
6
9
xj ³ 0, j 1, 2,,9
20
Alternative Formulation

2x9
min
z
3x2

x3

3x5
x6
x7

4x8





5y1 7y2 9y3


4

s.t.
2x1

x4

2x5
2x6
x7
y1

10,000





x2

x5

2x7
x8



y2

30,000





x3

x6

x8
2x9

y3

20,000





xj ³ 0, j 1,,9 yi ³ 0, i 1, 2, 3

where yi is overproduction of width i
21
5. Constrained Regression
Data (x,y) (1,2) , (3,4) , (4,7)
y

7
6
5

4
3

2
1
x
1 2 3 4 5
We want to fit a linear function y ax b to
these data points i.e., we have to choose
optimal values for a and b.
22
Objective Find parameters a and b that minimize
the maximum absolute deviation between the data
yi and the fitted line yi axi b.
Ù
Ù
y
y
and

i
i

observed value
Predicted value
In addition, were going to impose a priori
knowledge that the
slope of the line must be positive. (We dont
know about the intercept.)
a slope of line
known to be positive
Decision variables
positive or nega
tive
b y-intercept



-
³
³
0, b-
b
b
b-
, b

0


23
Objective function
Ù




i 1, 2, 3
-
Let w
max

yi
yi


Ù
where yi axi b
Optimization model
w
Min
Ù



s.t.
³
-
w
yi
yi
i 1, 2, 3

24
Nonlinear constraints
Ù


w ³


a(1)
-
y
b 2
y

1
1
Ù
w ³


a(3)


b 4
-
y
y

2
2
Ù

w ³




-
b 7
a(4)
y
y

3
3
Convert absolute value terms to linear terms
Note 2 ³x iff 2 ³ x and 2 ³ -x
Ù



³

Thus
w
-
is equivalent to
y
y
i
i

w ³ axi b - yi and w ³ - axi b yi
25
Min
w
so finally
-
s.t.
-
-

a b


b

w
2
-

-
-

-
a - b
b

w

2
-
-
-

3a b


b

w
4
-
-
-
-

-
3a
b

b

w

4
-
-
-


4a b

b

w
7
-

-

-

-
-
4a
b

b

w

7
-
³

a, b
, b
, w
0