Linear Program
MAX
s.t.
CBXB + CNBXNB
BXB + ANBXNB = b
XB ,
XNB ≥ 0
Important LP Equations
XB  B b -1
Z  CB B b -1
B
-1
jNB
a jx j
 C B a
-1
jNB
B
j
- cj  x j
Important LP Derivatives
Z
-1
 - (C B B a j - c j )
x j
j  NB
X B
-1
 - B a j j  NB
x j
Duality
Pr imal
Max
s.t.
Dual
CX
Min
AX  b
X
0
s.t.
Ub
UA  C
U
0
Duality
Primal Solution
Item
Solution
Primal
function
Objective
Information
Shadow prices
Dual Solution Item
Corresponding Dual Solution Information
Objective function
Variable values
Slacks
Reduced costs
Variable values
Shadow prices
Reduced costs
Slacks
Unbounded Solution
x2
Objective
increases
x1
Infeasible Solution
x2
A
B
x1
Multiple Optima
x2
Isocline with
highest
objective
P1
P2
x1
Degeneracy
x2
x1
P1
Complementary Slackness
• derived from duality
*

 0
 U A  C X
*
 0
U
*'
 b  AX
*'
Reduced
Cost
-1
CBB a  - c
• Negative derivative of objective function
with respect to a variable
• At optimality:
– Zero for all basic variables
– Non-negative for all non-basic variables (max)
– Non-positive for all non-basic variables (max)
Multi-input, Multi-output
Max
c X
p
p
-
Xp
j
j
-
-
q
pj
e Z
k
k
k
j
p
s.t.
d Y
Yj

0

0
j
r Y
kj
j
mj
Yj
-
Zk
j
s
 bm
j
Xp
,
Yj
,
Zk

0
Mixing / Blending
Min
c F
a F
a F
F
j
j
j
s.t.
ij j
 UL i
for all i
ij j

LL i
for all i
j

1

0
j
j
j
Fj
for all j
Spatial Equilibrium (GAMS Ex.)
Max
 pr,k Yr,k

T
c
 r,r,k Tr,r,k

r, j,u
r,r,k
r,k
X
c
 r, j,u Xr, j,u
a
s.t.
r, j,u,i
X r, j,u

b r,i
X r, j,u

0
j
  Tr,r,k
r
Yr,k
 Tr,r,k

y
r, j,u,k
j,u
r
 d r,k
Yr,k
Yr,k
,
T
,
X r, j,u

0
Sequencing
Max - c jX jt1
 X jt1
t1 t
j
  a jX jt
j
X jt ,
 d Y

kt 2
k

 e Z
s
st 3
t3
s
t2
k
t1
j
s.t.
-
Y

0
st 3

0
st
 g mt
kt 2
k t2 t
  Ykt 2

t3 t
s
k t2 t
b Y
j
 Z
kt

f Z
s
s
k
Ykt 2 ,
Zst 2

0
Sequencing
Week1 - X1
Week2 - X1
-
X2
Week3 - X1
-
X2
-
X3
Week3
Y1

Y1

Y2

Y1

Y2

Y3
 dY1
Week1 aX 1
Week2

cX 3
0

0

0
 T1
 eY2
bX 2

 T2
 fY 3
 T3
Storage
Max
c X
t
t
-
t
s.t.
 cs H
t
t
t
t T
X1
Xt
-
H t-1
XT
-
H T-1
 H1 
s0
 Ht

0

0
Xt
 Ut
Xt

Lt

0
Xt ,
Ht
Lexicographic preferences
Min
s.t.
wi
wr 
 Tr
glr
glr   g rjX j  0
for all r
for all r
j
a
mj
X j  bm
for all m
j
wr
 wr
for all r  i
wr

for all r  i
wr
0
for all r
Xj  0
for all j
glr
unrestricted for all r
Weighted Preferences
Max
c q
r
r
r
  g rjX j  gl r  0
s.t.
for all r
j
Nrqr
a
 glr  0
mj
Xj
for all r
 b m for all m
j
qr
Xj
0
for all r
0
for all j
glr unrestricted for all r
Cost
Well behaved, Separable Function
Input X
A1
A2
A3 A4
Cost
Well behaved, Separable Function
c4
c3
c2
Input X
c1
A1
A2
A3 A4
Well behaved, Separable Function
Min
c S
S
i
i
i
s.t.
i
 X 
0
i
 di
Si
Si
,
X 
0
for alli
Disequilibrium – Known Life
Max
t
(1

r)
C jt X j,t

t
s.t.
 (1  r)  T
j
F I
je je
j
A
j e K j
ije
e
eK j
X j,t e

X j,e
 X*j,e
 X j,T e

bit
I je

0
I je

0
j
X j,t ,
Disequilibrium – Unknown Life
Max
  (1  r)
t
s.t.
j eK j
t
 (1  r)  T
F
C je
X j,t,e
A
X j,t,e
 bit
X j,0,e
 X*j,0,e
j e K j
ije
X j,T,e
X j,t 1,e 1
j eK j

je
I je
I je
 X j,t,e
X j,t,e
 0
 0
,
I je
 0
Equilibrium - Unknown Life
Max
 C
j
s.t.
X je
ije
X je
e
 A
j
je
 bi
e
X je
X j,e 1 
0
X je

0
Fixed Costs
Max CX
s.t.
X
-
FY
MY  0
 0
X
Y
 0 or 1
Fixed Capacity
Max
C
m
m
s.t.
- Fk Yk
Xm
k
Xm
  Cap km Yk

0

0
k
Xm
Yk
 0 or 1
Minimum Habitat Size
population
hmin
0
HAB0
area
HAB1
Minimum Habitat Size
HAB
HAB
h
HAB0 ,
min
I 
0
HAB1
  h max  h min   I 
0
 d  HAB1
 d  h min  I 
0
HAB1

0
0
POP
POP,
 h
0
min
I 
0,1
Warehouse
F V
Min
k
k
k
s.t.

 C
i
k
ik
X ik
X

 D
k
ik
kj
Ykj
j
Y
Y
kj
k

i
 E Z
ij
ij
Z
ij

ij
 Dj
i

k
 X ik

kj
j
Si
j

Z
i

0

0
j
CAPk Vk

Y
kj
j

A mk Vk
 bm
k
Vk  0 or 1,
X ik ,
Ykj ,
Zij 
0
Mutual exclusive products
 MY1
X
Z
Y1
 0
 MY2
 0

 1
Y2
 0
X, Z
Y1
,
Y2
 0 or 1
Either-Or-Active constraints
A1X
-
MY 
b1
A 2 X + MY  b 2  M
X
Y

0

0 or 1
Distinct Variable Values
X -V1Y1
-V2 Y2
Y1
 Y2
X
... -Vk Yk

0
 Yk

1

0
...
...
Y1 ,
Y2 , ...
Yk
 0 or 1
Badly behaved non-linear functions
Badly behaved non-linear functions
1
 2
 3
 4
1
-
-
 Z2
 Z3
Z1
Z2
,
0

0
Z4

0
 Z4

2

1
 Z4

1
 Z4

1

0
 Z3
Z1
3

-
Z1
,
0
Z3
4
2

Z2
3
,
1
Z1
2
1

4
Z1
,
Z2
,
Z3
,
Z4
= 0 or 1
Non-linear Programming
• Specification often straightforward
• Solving more difficult
– scaling (manual vs. computer)
– lower bounds to avoid division by zero and
other illegal operations
– local versus global extremes