CC oo nn tt ii nn uu oo uu ss LL ii nn ee aa rr PP rr oo gg rr aa mm mm ii nn gg ..SS ee pp aa rr aa tt ee dd CC oo nn tt ii nn uu oo uu ss LL ii nn ee aa rr PP rr oo gg rr aa mm mm ii nn gg

Bellman (1953)

max ( ) ( )

( ) ( ) ( , ) ( ) ( )

( ) ,

′

+ ≤

≥ < <

∫

∫

c t u t dt

H t u t G s t u s ds a t

u t t T

T

t

0

0

0 0

CLP

(Dantzig, Tyndall, Grinold, Perold, Anstreicher 60's-80's)

Anderson (1978)

max ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

, , ,

′

+ =

+ =

≥ < <

∫

∫

c t u t dt

Gu s ds x t a t

H u t z t b t

x u z t T

T

t

0

0

0 0

SCLP

(Anderson, Nash, Philpott, Pullan 80's-90's).

Duality, Structure, Solution by Discretization.

In 2000, W. proposed a simplex algorithm to solve:

max ( ) ( )

( ) ( )

( ) ( )

, , ,

T t c u t dt

Gu s ds x t a t

H u t z t b

x u z t T

T

t

− ′

+ = +

+ =

≥ < <

∫

∫

0

0

0 0

α SCLPwith linear data

MS&E324, Stanford University, Spring 2002 6B-1 Gideon Weiss© manufacturing & control

PP uu ll ll aa nn ’’ ss DD uu aa ll FF oo rr mm uu ll aa tt ii oo nn ::

Primal Problem:

max ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

, , ,

c T t u t dt

Gu s ds x t a t

H u t z t b t

u x z t T

T

t

− ′

+ =

+ =

≥ < <

∫

∫

0

0

0 0

Pullan s SCLP©

Dual Problem

min ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) , , , ,

a T t d t b T t r t dt

G t H r t q t ct

r q t T

T T− ′ + − ′

′ + ′ − =

= ↑ ≥ < <

∫ ∫π

π

π π

0 0

0 0 0 0

Pullan s SCLP© *

Theorem (Pullan)Assume H u t b t u t( ) ( ), ( )≤ ≥ 0 has bounded feasible region.• For a b c, , piecewise analytic, there is a solution which is piecewise analytic,• For a c, piecewise polynomial of degree n +1,b piecewise polynomial of degree nthere is a solution with u t( ) piecewise polynomial of degree n.• For a c, piecewise linear, b piecewise constant, there is a solution with u t( ) piecewise constant.• Number of breakpoints finite bounded (not polynomial)•• Strong duality holds

Pullan’s Algorithm: Sequence of discretizations, converging (in infinitenumber of steps) to the optimal solution.

Note: Separated is very different from General:in the one x t u s ds( ) ( )= ∫ , direct integrationin the other u t u s ds( ) ( )= +∫ K differential equations

MS&E324, Stanford University, Spring 2002 6B-2 Gideon Weiss© manufacturing & control

SS CC LL PP ww ii tt hh SS yy mm mm ee tt rr ii cc PP rr ii mm aa ll // DD uu aa ll

max ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

, ,

c T t u t dt d T t y t dt

Gu s ds Fy t x t a t

H u t b t

x u t T

SCLP

T T

t

− ′ + − ′

+ + =

=

≥ < <

∫ ∫

∫

0 0

0

0 0

min ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

, ,

*

a T t p t dt b T t r t dt

G p s ds H r t q t c t

F p t d t

p q t T

SCLP

T T

t

− ′ + − ′

′ + ′ − =

′ =

≥ < <

∫ ∫

∫

0 0

0

0 0

Here: u p, are primal and dual controls (≥ 0),x q, are primal and dual states (≥ 0),y r, are primal and dual supplementary (unrestricted)

Dimensions: G K J H I J F K L× × ×, ,

MS&E324, Stanford University, Spring 2002 6B-3 Gideon Weiss© manufacturing & control

WW ee aa kk DD uu aa ll ii tt yy aa nn dd CC oo mm pp ll ee mm ee nn tt aa rr yy SS ll aa cc kk nn ee ss ss

Dual Objective = a T t p t dt b T t r t dt

u s G ds y T t F p t u T t H r t dt

u T t G p s ds H r t dt y T

T T

TT t

Tt

( ) ( ) ( ) ( )

{ ( ) ( ) } ( ) ( ) ( )

( ) { ( ) ( )} (

− ′ + − ′

≥ ′ ′ + − ′ ′ + − ′ ′

= − ′ ′ + ′ +

∫ ∫

∫ ∫

∫ ∫

−

0 0

00

00 −− ′ ′

≥ − ′ + − ′ =

∫

∫ ∫

t F p t dt

u T t c t dt y T t d t dt

T

T T

) ( )

( ) ( ) ( ) ( )

0

0 0Primal Objective

Equality will hold if and only if

0 00 0

T Tx T t p t dt u T t q t dt∫ ∫− ′ = − ′ =( ) ( ) , ( ) ( ) .

Complementary Slackness: almost everywherex t p T t

u t q T t

( ) ( )

( ) ( )

> ⇒ − =

> ⇒ − =

0 0

0 0Corollary: The following are equivalent(a) u x y p q r, , , , are complementary slack feasible primaland dual solutions(b) they are optimal and have the same objective value (noduality gap)

(Strong duality no duality gap - exists in LP but not in other

MS&E324, Stanford University, Spring 2002 6B-4 Gideon Weiss© manufacturing & control

problems, unless additional conditions are imposed).

BB oo uu nn dd aa rr yy VV aa ll uu ee ss

For infinitesimal values of T we solve:max ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

, ,

c T t u t d T t y t

Fy t x t a t

H u t b t

x u t T

− ′ + − ′

+ =

=

≥ < <0 0

min ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

, ,

a T t p t b T t r t

H r t q t c t

F p t d t

p q t T

− ′ + − ′

′ − =

′ =

≥ < <0 0

These separate into 2 sets of primal and dual problems:max ( ) ( )

( ) ( ) ( )

( ) ,

d T y

Fy x a

x

boundary LP

′

+ =

≥

0

0 0 0

0 0

min ( ) ( )

( ) ( ) ( )

( ) ,

*b T r

H r q c

q

boundary LP

′

′ − =

≥

0

0 0 0

0 0

Assumption (Feasibility and Boundedness):The boundary problems are feasible and bounded.

MS&E324, Stanford University, Spring 2002 6B-5 Gideon Weiss© manufacturing & control

⇒ SCLP is feasible and bounded for some range of T

TT hh ee AA ss ss oo cc ii aa tt ee dd LL PP

Our solution of SCLP will evolve through solution of thefollowing LP / LP*, under varying sign restrictions.

max «( ) ( ) ( ) «( )

( ) «( ) ( ) «( )

( ) ( )

( ) , «( ) " ",

c T t u t d T t y t

G u t Fy t t a t

H u t b t

u t y t U t T

LP

− ′ + − ′

+ + =

=

≥ < <

ξ

0 0

min «( ) ( ) ( ) «( )

( ) «( ) ( ) «( )

( ) ( )

, «" ",

*

a t p T t b t r T t

G p T t H r T t T t c T t

F p T t d T t

p r U t T

LP

′ − + ′ −

′ − + ′ − + − = −

′ − = −

≥ < <

θ

0 0

Note: the time indices on the variables are only labels,we label the variables to indicate their role:

ξ θ= =«, «, «, «x y q r

Assumption (Non-Degeneracy):The r.h.s. of LP / LP* is (a.e. t) in general position to

column spaces ofG F I

H 0 0

,

′ ′

′

G H I

F 0 0

⇒ SCLP is non-degenerate: Solution is unique and at an

MS&E324, Stanford University, Spring 2002 6B-6 Gideon Weiss© manufacturing & control

extreme point in the function space of the solutions.

TT hh ee SS tt rr uu cc tt uu rr ee TT hh ee oo rr ee mm

Consider any functionsu t x t y t p t q t r t t T( ), ( ), ( ), ( ), ( ), ( ), 0 < < which satisfy

• u p x y q r, , , , ,≥ 0 are absolutely continuous.

Hence x y q r, , , have derivatives a.e.

• x y( ), ( )0 0 solve the boundary-LP,q r( ), ( )0 0 solve the boundary-LP*.

For x y, these are values at the left boundary,q r, run in reversed time so this is at the right boundary.

• At all times t , the rate functions (derivatives)u t t y t p T t T t r T t t T( ), ( ), «( ), ( ), ( ), «( ),ξ θ− − − < <0are complementary slack feasible solutions for LP / LP*.

Thus they are optimal solutions for LP / LP* under somesign configuration of ξ θ, . By non-degeneracy they are basicand unique solutions.

• At all times t , if x t tk k( ) , ( )> 0 ξ is basic, ifq T t T tj j( ) , ( )− > −0 θ is basic.

• x t q t t T( ) ( ) ,≥ ≥ < <0 0 0 .

Theorem: A solution to SCLP / SCLP* is optimal if (and

MS&E324, Stanford University, Spring 2002 6B-7 Gideon Weiss© manufacturing & control

only if for linear data) it has these properties.

RR ee ss tt rr ii cc tt ii oo nn tt oo tt hh ee LL ii nn ee aa rr DD aa tt aa CC aa ss eeProblem:max ( ( ) ) ( ) ( )

( ) ( ) ( )

( )

, ,

( )

′ + − ′ + ′

+ + = +

=

≥ < <

∫ ∫

∫

γ

α

T t c u t dt d y t dt

Gu s ds Fy t x t at

H u t b

x u t T

SCLP

linear data

T T

t

0 0

0

0 0

min ( ( ) ) ( ) ( )

( ) ( ) ( )

( )

, ,

( )

*

α

γ

+ − ′ + ′

′ + ′ − = +

′ =

≥ < <

∫ ∫

∫

T t a p t dt b r t dt

G p s ds H r t q t ct

F p t d

p q t T

SCLP

linear data

T T

t

0 0

0

0 0Boundary:

max

,

min

,

′

+ =

≥

′

′ − =

≥

d y

Fy x

x

b r

H r q

q

N

N N

N

0

0 0

0 0 0

α γ

Associated LP

MS&E324, Stanford University, Spring 2002 6B-8 Gideon Weiss© manufacturing & control

max «

«

, « " ",

min «

«

, «" "

′ + ′

+ + =

=

≥

′ + ′

′ + ′ − =

′ =

≥

c u d y

G u Fy at

H u b

u y U

a p b r

G p H r c

F p d

p r U

ξ θ

0 0

TT hh ee AA ll gg oo rr ii tt hh mm

The algorithm constructs a solution for SCLP / SCLP* forthe whole range of time horizons, 0 < < ∞T .

Corollary 1: The solution for any time horizon T with theexception of a finite list: 0 0 1= < < < ∞T T T R( ) ( ) ( )L , is characterized by a sequence of adjacent bases of LP, B BN1, ,K .

For a particular T there are time breakpoints 0 0 1= < < =t t t TNL such that for t t tn n− < <1 ,

u y p rn n n n n n, , « , , , «ξ θ are basic Bn.

In the single pivot B Bn n→ +1 let v leave the basisif v k= ξ then we have equation x tk n( ) = 0.if v uj= then we have equation q T tj n( )− = 0.

Corollary 2: The solution of the equations x tk n( ) = 0,q T tj n( )− = 0, together with:

τ τ τ1 1+ + = = − −L N n n nT t t( )determines the solution for a whole range of time horizons,T T Tr r( ) ( )− < < < ∞1 .

MS&E324, Stanford University, Spring 2002 6B-9 Gideon Weiss© manufacturing & control

AA ll gg oo rr ii tt hh mm ii cc SS tt rr uu cc tt uu rr ee oo ff SS oo ll uu tt ii oo nn ::

For time horizon T , Break points: 0 0 1= < < < =t t t TNLτn n nt t= − −1

N −1 Equations:

Adjacent B B vn n n→ =+1 : leaves:

v x t x

v u q T t q

n k k n k km

m

n

m

n j j n jN

jm

m n

N

m

= ⇒ = + =

= ⇒ − = + =

=

= +

∑

∑

ξ ξ τ

θ τ

( ) ,

( ) .

0

1

1

0

0

and equation N : τ τ1 + + =L N T

Slack Inequalities, at local minima of x qk j, :

ξ ξ ξ τ

θ θ θ τ

kn

kn

k n k km

m

n

m

jn

jn

j n jN

jm

m n

N

m

n N x t x

n q T t q

< > = ⇒ = + ≥

= > < ⇒ − = + ≥

+

=

+

= +

∑

∑

0 0 0

0 0 0 0

1 0

1

1

1

, ( ) ,

, ( ) .

or

or

MS&E324, Stanford University, Spring 2002 6B-10 Gideon Weiss© manufacturing & control

Algorithm:

Step 1: Solution for T T T( ) ( )0 1< < is given by the solution of boundary-LP / boundary-LP*.

Step r: Extend the solution to next time horizon range.

1 1 0

0

L

A

B I

T

d

g

r

=

+

τ

σ

( ) ∆

Step R+1 Stop when next ∆ can be ∞,or when solution infeasible.

MS&E324, Stanford University, Spring 2002 6B-11 Gideon Weiss© manufacturing & control

AA ll gg oo rr ii tt hh mm CC aa ss ee ii ii ii

Buffer empties at T

T<T (l)

T=T (l)

T>T (l)

ξ leaves

u enters

x 1

x 2

1

2

x 1

x 2

x 2

x 1

MS&E324, Stanford University, Spring 2002 6B-12 Gideon Weiss© manufacturing & control

AA ll gg oo rr ii tt hh mm CC aa ss ee ii ii

Interval shrinks at tn

x 1 q 3

B'→C ξ leaves

C→B'' u leaves

1

3

T<T (l)

x 1 q 3

B'→B'' ξ u leave1 3

T=T (l)

x 1 q 3

B'→D u leaves

D→B'' ξ leaves

3

1

T>T (l)

B'

B'

B'

B''

B''

B''

C

D

MS&E324, Stanford University, Spring 2002 6B-13 Gideon Weiss© manufacturing & control

AA ll gg oo rr ii tt hh mm CC aa ss ee ii ii ,, SS uu bb pp rr oo bb ll ee mm

Interval shrinks at tn : subproblem for additional intervals

x 1 q 3

B'→C ξ leaves

C→B'' u leaves

1

3

T<T (l)

x 1 q 3

B'→B'' ξ u leave1 3

T=T (l)

B'

B'

B''

B''

C

x 1 q 3

→D u leaves

→B'' ξ leaves

3

1

T>T (l)

B' B''D

x 1 q 3

B'→B'' ξ u leave1 3

T=T (l)

B' B''

D EE1 2

E1 E2

E1

E2

subproblem

Subproblem solved by recursive call to the algorithm, with smaller,modified problem.

MS&E324, Stanford University, Spring 2002 6B-14 Gideon Weiss© manufacturing & control

EE xx aa mm pp ll ee 11 ::Pullan solved the following ASSET DISPOSAL PROBLEM in his 1993 paper:

G H

a t t a t t

b t c t c t t t

=

= [ ]

= + = +

= = = − < <

1 0

0 11 2

4 3 2

10 2 0 21 2

1 1 2

, ,

( ) , ( ) ,

( ) , ( ) ( ) , .

max ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

, , ,

2 2

4

3 2

10

0 0

1 202

1 10

2 20

1 2 1

− + −

+ = +

+ = +

+ + =

≥ < <

∫

∫

∫

t u t t u t dt

u s ds x t t

u s ds x t t

u t u t z t

u x z t T

t

t

Our formulation, for arbitrary time horizon is:

max ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

, , ,

T t u t T t u t dt

u s ds x t t

u s ds x t t

u t u t z t

u x z t T

T

t

t

− + −

+ = +

+ = +

+ + =

≥ < <

∫

∫

∫

1 20

1 10

2 20

1 2 1

4

3 2

10

0 0

MS&E324, Stanford University, Spring 2002 6B-15 Gideon Weiss© manufacturing & control

EE xx aa mm pp ll ee 11 ::

The optimal solution is:

12

It is solved in 3 steps:

For time horizons 0 49

< <T : Empty buffer 1 only,

u t x t t x t t1 1 210 4 9 3 2( ) , ( ) , ( ) .= = − = +

For time horizons 49

2< <T : keep buffer 1 empty, and empty buffer 2:

u t x t u t x t t1 1 2 21 0 592

52

( ) , ( ) , ( ) , ( )= = = = −

For time horizons 2 < < ∞T : keep buffers 1 and 2 empty:

u t x t u t x t1 1 2 21 0 2 0( ) , ( ) , ( ) , ( )= = = =

NOW WATCH PULLAN SOLVE IT:

MS&E324, Stanford University, Spring 2002 6B-16 Gideon Weiss© manufacturing & control

MS&E324, Stanford University, Spring 2002 6B-17 Gideon Weiss© manufacturing & control

EE xx aa mm pp ll ee 22 ::

A 2 machine 3 buffer example, m m m2 1 3> + :

m1

1m2

m3

2

3

m > m + 2 1 3

Variation

1

3

2

m

MS&E324, Stanford University, Spring 2002 6B-18 Gideon Weiss© manufacturing & control

(Solution of this example is joint with Florin Foram, 1992)

SS oo ll uu tt ii oo nn

0 1

1 1 2 3 3 2

≤ ≤

=

T T

B u z

( ),

{ , , , , }

Buffer 3 drains

ξ ξ ξ

T T T

B u u z

B u u

( ) ( ),

{ , , , , }

{ , , , , }

1 2

3 1 2 2 3 1

2 1 2 2 3 3

≤ ≤

=

=

Buffer 2 drains

Optimal basis

Subproblem

ξ ξ

ξ ξ ξ

T T T

B u u u z

( ) ( ),

{ , , , , }

2 3

4 1 1 2 3 1

≤ ≤

=

Flow out of buffer 1

until interval 3 shrinks

ξ

T T T

B

B u u u

( ) ( ),

© { , , , , }

3 4

3

3 1 1 2 3 3

≤ ≤

=

Basis repalced by

ξ ξ

MS&E324, Stanford University, Spring 2002 6B-19 Gideon Weiss© manufacturing & control

T T

B u u u z z

( ) ,

{ , , , , }

4

5 1 2 3 1 2

≤ ≤ ∞

=

empty

EE xx aa mm pp ll ee 33 ::55 mm aa cc hh ii nn ee 22 00 bb uu ff ff ee rr rr ee -- ee nn tt rr aa nn tt ee xx aa mm pp ll ee

12

3

54

76

911

12

14

16

17

8

10

13

1819

20

15

I II III IV V

MS&E324, Stanford University, Spring 2002 6B-20 Gideon Weiss© manufacturing & control

SS oo ll uu tt ii oo nn ::

MS&E324, Stanford University, Spring 2002 6B-21 Gideon Weiss© manufacturing & control

EE xx aa mm pp ll ee 44 ::

A continuous time Leontief System

MS&E324, Stanford University, Spring 2002 6B-22 Gideon Weiss© manufacturing & control1 2 3 4 5

X

0

50

100

150

200

250

1 2 3 4 5

X

0

50

100

150

200

250

EE vv oo ll uu tt ii oo nn oo ff tt hh ee ss oo ll uu tt ii oo nn ::

MS&E324, Stanford University, Spring 2002 6B-23 Gideon Weiss© manufacturing & control

x4

x4

q6

r2x3

q6

r2x3

x2

q12

r2

x3

x4

t

T