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