linear program
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Linear Program
MAX CBXB + CNBXNB
s.t. BXB + ANBXNB = b
XB , XNB ≥ 0
Important LP Equations
NBj
jj-1-1
B xaB - bB X
-1 -1B B j j j
j NB
Z C B b - C B a - c x
Important LP Derivatives
NBj )c - aB(C- x
Zjj
1-B
j
NBj aB- x
Xj
1-
j
B
Duality
Pr imal Dual
Max CX Min U b
s.t. AX b s.t. U A C
X 0 U 0
Duality
Primal Solution Item Primal Solution Information
Dual Solution Item Corresponding Dual Solution Information
Objective function Objective function Shadow prices Variable values
Slacks Reduced costs
Variable values Shadow prices
Reduced costs Slacks
Unbounded Solution
Objective increases
x1
x2
Infeasible Solution
x1
x2
A B
Multiple Optima
x1
x2 P1
P2
Isocline with highest objective
Degeneracy
x1
x2
P1
Complementary Slackness
• derived from duality
*' *
*' *
0U b AX
0U A C X
Reduced Cost
• 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)
-1BC B a - c
Multi-input, Multi-output
p p j j k kp j k
p pj jj
kj j kj
mj j mj
p j k
Max c X - d Y - e Z
s.t. X - q Y 0
r Y - Z 0
s Y b
X , Y , Z 0
Mixing / Blending
j allfor 0F
1F
i allfor LLFa
i allfor ULFas.t.
FcMin
j
jj
ij
jij
ij
jij
jjj
T Xr,k r,k r,r ,k r,r,k r, j,u r, j,u
r,k r,r ,k r, j,u
r, j,u,i r, j,u r,ij
r,r ,kr
r,k r, j,u,k r, j,uj,ur,r,k
r
r,k r,k
r,k r, j,u
Max p Y c T c X
s.t. a X b
TY y X 0
T
Y d
Y , T , X 0
Spatial Equilibrium (GAMS Ex.)
Sequencing
1 2 3
1 2 3
1 2
1 2
2 3
2 3
2 2
j jt k kt s stj t k t s t
jt ktj t t k t t
kt stk t t s t t
j jt j kt s st mtj k s
jt kt , st
Max - c X - d Y e Z
s.t. X Y 0
Y Z 0
a X b Y f Z g
X , Y Z 0
Sequencing
333
222
111
321321
2121
11
TfYcXWeek3
TeYbXWeek2
TdYaXWeek1
0YYY X-X -X-Week3
0YY X -X-Week2
0Y X-Week1
Storage
t t t tt t
t T
1 1 0
t t-1 t
T T-1
t t
t t
t t
Max c X - cs H
s.t. X H s
X - H H 0
X - H 0
X U
X L
X , H 0
Lexicographic preferences
i
r r r
r rj jj
mj j mj
r r
r
r
j
r
Min w
s.t. w gl T for all r
gl g X 0 for all r
a X b for all m
w w for all r i
w for all r i
w 0 for all r
X 0 for all j
gl unrestricted for all r
Weighted Preferences
r rr
rj j rj
r r r
mj j mj
r
j
r
Max c q
s.t. g X gl 0 for all r
N q gl 0 for all r
a X b for all m
q 0 for all r
X 0 for all j
gl unrestricted for all r
Well behaved, Separable Function
A1 A2 A3 A4
Cos
t
Input X
Well behaved, Separable Function
A1 A2 A3 A4
Cos
t
Input X c1
c4
c3
c2
Well behaved, Separable Function
i ii
ii
i i
i
Min c S
s.t. S X 0
S d for all i
S , X 0
Disequilibrium – Known Life
j
j
t Tjt j,t je je
t j j ee K
ije j,t e itj e K
*j, e j, e
j,T e jej
j,t je
Max (1 r) C X (1 r) F I
s.t. A X b
X X
X I 0
X , I 0
Disequilibrium – Unknown Life
j j
j
t Tje j,t,e je je
t j e K j e K
ije j,t,e itj e K
*j,0,e j,0,e
j,T,e je
j,t 1,e 1 j,t,e
j,t,e je
Max (1 r) C X (1 r) F I
s.t. A X b
X X
X I 0
X X 0
X , I 0
Equilibrium ‑ Unknown Life
je jej e
ije je ij e
je j,e 1
je
Max C X
s.t. A X b
X X 0
X 0
Fixed Costs
Max CX - FY
s.t. X - MY 0
X 0
Y 0 or 1
Fixed Capacity
m m k km k
m km kk
m
k
Max C X - F Y
s.t. X Cap Y 0
X 0
Y 0 or 1
Minimum Habitat Size
hmin 0
area
population
HAB0 HAB1
Minimum Habitat Size
0 min
0 min
1 max min
1 min
0 1
HAB h
HAB h I 0
HAB h h I 0
POP d HAB d h I 0
POP, HAB , HAB 0
I 0,1
Warehouse
k k ik ik kj kj ij ijk i k k j i j
ik ij ik j
kj ij jk i
ik kji j
k k kjj
mk k
mk
k ik kj ij
Min F V C X D Y E Z
s.t. X Z S
Y Z D
X Y 0
CAP V Y 0
A V
b
V 0 or 1, X , Y , Z 0
Mutual exclusive products
1
2
1 2
1 2
X MY 0
Z MY 0
Y Y 1
X, Z 0
Y , Y 0 or 1
Either-Or-Active constraints
1 1
2 2
A X - MY b
A X + MY b M
X 0
Y 0 or 1
Distinct Variable Values
1 1 2 2 k k
1 2 k
1 2 k
X -V Y -V Y ... -V Y 0
Y Y ... Y 1
X ... 0
Y , Y , ... Y 0 or 1
Badly behaved non-linear functions
A1 A2 A3 A4
Cos
t
Input X
Badly behaved non-linear functions
1 2 3 4
1 1
2 2
3 3
4 4
1 2 3 4
1 3
1 4
2 4
1 2 3 4
1 2 3 4
1
- Z 0
- Z 0
- Z 0
- Z 0
Z Z Z Z 2
Z Z 1
Z Z 1
Z Z 1
, , , 0
Z , Z , Z , Z = 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
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