Short-Term Scheduling under Uncertainty Marianthi Ierapetritou Department Chemical and Biochemical Engineering Piscataway, NJ 08854-8058.

Short-Term Scheduling under Uncertainty

Marianthi Ierapetritou

Department Chemical and Biochemical Engineering

Piscataway, NJ 08854-8058

Sandia National LaboratoriesLarge-Scale Robust Optimization

Objective

Identify and reduce bottlenecks at different levels

Integration of the whole decision-making process

Online Control

Short-term Scheduling

Production Planning

Supply Chain Management

Time Horizon

Uncertainty Complexity

Opportunity for

Optimization

Process Operations Decision Making


Short-term scheduling Uncertainty in product prices, product demands, raw material

availability, machine availability, processing times

Production Planning Longer time horizon under consideration (several months) Larger number of materials and products Uncertainty in facility availability, product demands, orders, raw

materials

Supply chain management Multiple sites involving production, inventory management,

transportation Longer planning time horizon (couple of years) Uncertainty in material availability, costs, transportation

Uncertain Parameters


Process Plant Optimal Schedule

Given: Determine:Raw Materials, Required Products, Task Sequence, Production Recipe, Unit Capacity Exact Amounts of

materialProcessed

Scheduling objectives : Economic Maximize Profit, Minimize Operating Costs,

Minimize Inventory CostsTime Based Minimize Makespan, Minimize Tardiness

Short-term Scheduling


Binary variables to allocate tasks to resources

Continuous variables to represent timing and material variables

Mixed Integer Linear Programming Models

Smaller models that are computationally efficient and tractable

T2T2T1 T1

T1 T2 T2 T1 T2 T2 T2T1

Discrete time formulation

“Real” time schedule

Continuous time formulation

Event Points: When a tasks begins

Continuous Time Formulation


Deterministic Scheduling Formulation

minimize H or maximize price(s)d(s,n)

subject to wv(i,j,n) 1

st(s,n) = st(s,n-1) – d(s,n) + Pb(i,j,n-1) +

cb(i,j,n)

st(s,n) stmax(s)

Vmin(i,j)wv(i,j,n) b(i,j,n) Vmax(i,j)wv(i,j,n)

d(s,n) r(s)

Tf(i,j,n) = Ts(i,j,n) + (i,j)wv(i,j,n) + (i,j)b(i,j,n)

Ts(i,j,n+1) Tf(i,j,n) – U(1-wv(i,j,n))

Ts(i,j,n) Tf(i’,j,n) – U(1-wv(i’,j,n))

Ts(i,j,n) Tf(i’,j’,n) – U(1-wv(i’,j’,n))

Ts(i,j,n) H, Tf(i,j,n) H

Duration Constraints

Demand Constraints

Allocation Constraints

Capacity Constraints

Material Balances

Objective Function

n

s

(i,j)

M.G.Ierapetritou and C.A.Floudas. Effective continuous-time formulation for short-term scheduling. 1. Multipurpose batch processes. 1998


Increased Complexity: Parameter Fluctuations


Price of P1 is an uncertain parameter. Considering time horizon of 16 hours, $1 increase results in the following different production schedules.

Uncertainty impacts the optimal schedule

Uncertainty in Short-Term Scheduling


0 82 4 61 3 5 7

0 82 4 61 3 5 7

50.00

50.00

10.40

64.60 10.03 50.93

64.04 4.63 50.93

55.56

74.07 4.63 50.93

44.44 74.07 50.93

55.56 Deterministic Schedule

Deterministic Schedule

Robust ScheduleRobust

Schedule

heating

heating

reaction 1 reaction 2 reaction 3 reaction 2

reaction 1 reaction 2 reaction 3

separation

reaction 2 reaction 2reaction 3

reaction 3reaction 1 reaction 1

separation

demand (product 2) = 50*(1 + 60%)

demand (product 2) = 50

Standard Deviation = 0.29

E(makespan) = 7.24hr

E(makespan) = 8.15hr

Standard Deviation = 2.63

Uncertainty in Short-Term Scheduling


Literature Review: Representative Publications

Reactive Scheduling

S.J.Honkomp, L.Mockus, and G.V.Reklaitis. A framework for schedule evaluation with processing uncertainty. Comput. Chem. Eng. 1999, 23, 595 J.P.Vin and M.G.Ierapetritou. A new approach for efficient rescheduling of multiproduct batch plants. Ind. Eng. Chem. Res., 2000, 39, 4228

Handles uncertainty by adjusting a schedule upon realization of the uncertain parameters or occurrence of unexpected events

Stochastic ProgrammingUncertainty is modeled through discrete or continuous probability functions J.R.Birge and M.A.H.Dempster. Stochastic programming approaches to stochastic scheduling. J. Global. Optim. 1996, 9, 417 J.Balasubramanian and I.E.Grossmann. A novel branch and bound algorithm for scheduling flowshop plants with uncertain processing times. Comput. Chem. Eng. 2002, 26, 41


Literature Review: Representative Publications

Robust Optimization

X.Lin, S.L.Janak, and C.A.Floudas. A new robust optimization approach for scheduling under uncertainty – I. bounded uncertainty. Comput. Chem. Eng. 2004, 28, 2109

Produces “robust” solutions that are immune against uncertainties

Fuzzy Programming

H.Ishibuchi, N.Yamamoto, T.Murata and Tanaka H. Genetic algorithms and neighborhood search algorithms for fuzzy flowshop scheduling problems . Fuzzy Sets Syst. 1994, 67, 81 J.Balasubramanian and I.E.Grossmann. Scheduling optimization under uncertainty- an alternative approach. Comput. Chem. Eng. 2003, 27, 469

Considers random parameters as fuzzy numbers and the constraints are treated as fuzzy sets

MILP Sensitivity Analysis

Z.Jia and M.G.Ierapetritou. Short-term Scheduling under Uncertainty Using MILP Sensitivity Analysis. Ind. Eng. Chem. Res. 2004, 43, 3782

Utilizes MILP sensitivity analysis methods to investigate the effects of uncertain parameters and provide a set of alternative schedules


Disruptive EventsRush Order ArrivalsOrder CancellationsMachine Breakdowns

Not much informatio

n is

available

REACTIVESCHEDULING

Parameter UncertaintyProcessing timesDemand of productsPrices

Information is available

PREVENTIVE SCHEDULING

Uncertainty in Scheduling


model robustness

solution robustness

Data perturbation

New alternative schedules

Deterministic schedule

LB/UB on objective function

MILP

sensitivity

analysis

framework

Robust

optimization

method

A set of solutions represent trade-off between various

objectives

Preventive Scheduling



Robust Optimization

MILP Sensitivity

Analysisminimize H or maximize price(s)d(s,n)

subject to wv(i,j,n) 1st(s,n) = st(s,n-1) – d(s,n) + Pb(i,j,n-1) + cb(i,j,n)

st(s,n) stmax(s) Vmin(i,j)wv(i,j,n) b(i,j,n) Vmax(i,j)wv(i,j,n)

d(s,n) r(s) Tf(i,j,n) = Ts(i,j,n) + (i,j)wv(i,j,n) + (i,j)b(i,j,n)

Ts(i,j,n+1) Tf(i,j,n) – U(1-wv(i,j,n))Ts(i,j,n) Tf(i’,j,n) – U(1-wv(i’,j,n))Ts(i,j,n) Tf(i’,j’,n) – U(1-wv(i’,j’,n))

Ts(i,j,n) H, Tf(i,j,n) H

Mixed-Mixed-integer integer Linear Linear

ProgramminProgrammingg


mixing

reaction

separation

H (time horizon)100 2 4 8

55

55

50

mixing

reaction15

6

15

20separation

-Can the schedule accommodate the demand fluctuation?

-How the capacity of the units affect the production objective?

- What is the effect of processing time at the objective value?

Questions to Address


Inference-based MILP Sensitivity Analysis

iP Aijuj

P + sj(uj – uj) - iai rP

sjP i

PAij, sjP -qj

P, j = 1,…,n

rP = -qjPuj

P + Pa – zP +zP

cjujP - sj

P(ujP – uj

P) -rP

sjP -cj, sj

P -qjP, j = 1,…,n

qjP = i

PAij - iPcj

- for the perturbations - for the perturbations A and A and a a - for the perturbations - for the perturbations c c

Bound z z* - z holds if there are s1P,…,sn

P that satisfy:

minimize z = cxsubject to Ax a

0 x h, xj integer, j=1,…k

minimize z = (c + cc)xsubject to (A + AA)x a + aa

0 x h, xj integer, j=1,…k

Aim: Determine under what condition z z* - z remains valid

Partial assignment at node pxj {ujP,…,uj

P} j = 1,…,n

*M.W.Dawande and J.N.Hooker, 2000


Extract information from

the leaf nodes

Extract information from

the leaf nodes

Solve the deterministic scheduling problem using

B&B tree

Solve the deterministic scheduling problem using

B&B tree

Proposed Uncertainty Analysis Approach

- Range of objective

change for certain parameter change

- Range of objective

change for certain parameter change

Solve relaxed LP at the leaf nodes with perturbed data

Solve relaxed LP at the leaf nodes with perturbed data

Identify the feasible schedules by examining the

B&B tree

Identify the feasible schedules by examining the

B&B tree

Evaluate the alternative schedules

Evaluate the alternative schedules

- Robustness- Nominal performance- Average performance

- Robustness- Nominal performance- Average performance

MILP Sensitivity MILP Sensitivity AnalysisAnalysis


Obtain sequence of tasks

from original schedule

42

56

70

84

98

33 44 55 66 77

P1

P2

Generate random demands in expected

range

Makespan minimization is considered as the objective

Makespan to meet a particular demand is found using the sequence of tasks derived from original scheduleBinary variables corresponding to allocation of tasks are fixedBatch sizes and Starting and Finishing times of tasks are allowed to vary

Robustness Estimation


Inventory ofraw materialsintermediates

etc.

Hmax HinfROLLOVER

Meets maximum possibledemand

Meets unsatisfieddemand

Total makespan Hcorr = Hmax + HinfCorrected Standard Deviation:

Hact = Hp if scenario is feasible = Hcorr if the scenario is infeasible

totp

p tot

actcorr p

HHSD

1

2det

)1(

)(

Robustness under Infeasibility

J.P.Vin and M.G.Ierapetritou. Robust short-term scheduling of multiproduct batch plants under demand uncertainty. 2001


Case Study 1

S1S1 S2S2 S3S3 S4S4

wv(i1,j1,n0)

wv(i1,j1,n1)

wv(i2,j2,n1)

wv(i3,j3,n2)

wv(i2,j2,n2)

3.0

5.17 3.0

7.65 5.83 7.16infeasible

8.14 10.16 7.16

9.83infeasible

infeasible8.33

9.87 8.83

7.16

8.839.98

infeasible

9.83

(Schedule 1)

0

00

0

0 0 0 0

00

1

1

1

1 1

1 1

1

11

Effect of demand d~[20, 100]

-0.097 d Hdnom = 50

H’ Hnom + 0.097d = 12.73hd’ = 80

Hnom = 9.83h

mixingmixing reactionreaction purificationpurification

B&B tree with B&B tree with nominal nominal demanddemand


Case Study 1

(Schedule 2) (Schedule 3)

0

0 0

01

1 1

1(12.13)

schedule 1 schedule 2 schedule 3

Hnom(h)

Havg(h)

SDcorr

9.83

14.20

5.52

10.77

11.56

1.61

10.91

11.79

2.17

wv(i1,j1,n0)

wv(i1,j1,n1)

wv(i2,j2,n1)

wv(i3,j3,n2)

wv(i3,j3,n3)

wv(i2,j2,n2)

3.0

5.17 3.0

7.65 5.83 7.16infeasible

8.14 10.16 7.16

9.83infeasible

infeasible8.33

9.87 8.83

7.16

8.839.98infeasible

9.83

(Schedule 1)

0

00

0

0 0 0 0

00

1

1

1

1 1

1 1

1

11

(12.73)(17.97)

Schedule Schedule EvaluationEvaluation


5

7

9

11

13

15

20 30 40 50 60 70 80 90 100

Demand

H

schedule 1 (optimal when d ≤ 50)

schedule 2 (optimal when d ≥ 50)

schedule 3

Case Study 1


Case Study 1Effect of processing time T(i1,j1) ~ [2.0, 4.0]

profit’ profitnom + 24.48T = 47.04Tnom = 3.0

T’ = 4.0

profitnom = 71.52

schedule 1schedule 2 schedule 3

profitnom

profitavg

SDcorr

71.52

66.98

26.9

65.27

9.33

65.27

65.17

10.49

wv(i1,j1,n0)

wv(i1,j1,n1)

wv(i2,j2,n1)

wv(i3,j3,n2)

wv(i3,j3,n3)

wv(i2,j2,n2)

100

100 50

100

100 96.05

78.42

50

(75)

0

0

0

0

0

0

0

1

1

1

1

1

1

1

71.52

100

50

75

(Schedule 1)

50

75(75)

1 050

72.46(62.11)

78.42

64.61

(Schedule 2)

1 0

1

0

1

1

1

1 0

(Schedule 3)


Shortcoming of Proposed Approach

Since the entire analysis is based on a single tree among a large number of possible branch-and-bound trees that can be used to solve the MILP, it provides conservative sensitivity ranges.


Parametric Programming

z() = min cTx + dTysubject to Ax + Dy

bxL x xU

L U

x Rm, y (0,1)T

z() = min cTx + dTysubject to Ax + Dy - r b

cTx + dTy – z0 - = 0

yi - yi F1 - 1

xL x xU

L’ U’x Rm, y (0,1)T

iF1 iF 0

b = b0 + r

LP sensitivity analysis: z() = z0 +

L L’ U’ U

solved at b = b0+Lr

optimal solution (x*,y*)

Integer cut to exclude current optimal solution

b[b0+Lr, b0+Ur]

break point ’, new optimal solution (x*’,

y*’)

Fix integer variables at y*

A.Pertsinidis et al. Parametric optimization of MILP programs and a framework for the parametric optimization of MINLPs. 1998


Multiparametric MILP

J. Acevedo and E.N.Pistikopoulos. A Multiparametric Programming Approach for Linear Process Engineering Problems under Uncertainty. 1997

Select a branching variable

Select a branching variable

Solve the fully relaxed problem

Solve the fully relaxed problem

Solve the mpLP at the

nodes

Solve the mpLP at the

nodes

Compare the solution with the current UB, update the optimal function in the uncertain space

Compare the solution with the current UB, update the optimal function in the uncertain space

Based on simplex algorithm, check the neighboring bases of the LP

tableau

Based on simplex algorithm, check the neighboring bases of the LP

tableau


Shortcomings of Existing Approach

mpLP approach requires retrieving the LP

tableaus and visiting the neighbor bases

Solve mpLP at every node in the B&B tree during

the branch and bound procedure: can be a

computationally expensive effort


Proposed Analysis on the RHS for MILPs

minimize z = cxsubject to Ax a x 0, xj Є (0, 1), j=1,…k

minimize z = cxsubject to Ax a + ax 0, xj Є (0, 1) , j=1,…k

Develop a framework to investigate the effect of Δa on the optimal solution x and objective value z

A set of optimal integer solutions Critical regions Optimal functions


Proposed Approach: Single Uncertain Parameter

Find Δamax that leaves the structure of the

tree unchanged

Find Δamax that leaves the structure of the

tree unchanged

Solve the original problem at the nominal value using

a branch and bound method



Collect zp, λp at each leaf node

p

Collect zp, λp at each leaf node

p

For a = Δamax + ε update the B&B treeFor a = Δamax + ε

update the B&B tree


Δamax = min{Δabasis, min{ } }zp – z0

λp – λ0

Case 1: A descent node of the node 0

Case 2: A descent node of other leaf node

Case 3: Node 0, but the basis has changed

The new optimal node can be:

Determine Δamax and Update the B&B Tree

P

Update the B&B tree at a = Δamax +ε

where node 0 is the optimal node

0

*

case

1

*

case

2

case

3


Proposed Approach: Multiple Uncertain Parameters

mpLP at the leaf nodes

mpLP at the leaf nodes

mpLP algorithm

mpLP algorithm

Compare the critical regions with the current upper bounds Update the B&B tree

Compare the critical regions with the current upper bounds Update the B&B tree






mpLP Algorithm at the Leaf Nodes

maximize cx* – zsubject to z = max{z(k) + λ(k)θa + β(k)θb}

a0 ≤ θa ≤ a0 + Δa

b0 ≤ θb ≤ b0 + Δb

Bilevel Linear Programming

At each iteration, solve

where x* = argmin cxsubject to Ax ≥ θ

θ

cx*

max{z(k) + λ(k)θ}

cx*current optimal functions



maximize {min cx| Ax θ} – z

subject to z z(k) + λ(k)θa + β(k)θb

a0 ≤ θa ≤ a0 + Δa

b0 ≤ θb ≤ b0 + Δb

current optimal functions

check if there is any point at which max{z(k) + λ(k)θa + β(k)θb} is less than cx*

Stop when the objective = 0

Include an additional constraint z z(k+1) + λ(k+1)θa + β(k+1)θb to above problem


Solve the Bilevel Programming Problem

max θys.t. ATy ≤ c a0 ≤ θa ≤ a0 + Δa

b0 ≤ θb ≤ b0 + Δb

Y 0

min cxs.t. Ax θ a0 ≤ θa ≤ a0 + Δa

b0 ≤ θb ≤ b0 + Δb

x 0

Replace the inside optimization problem by its dual

max θy - zs.t. ATy ≤ cz z(k) + λ(k)θa + β(k)θb

a0 ≤ θa ≤ a0 + Δa

b0 ≤ θb ≤ b0 + Δb

Y 0

Single optimization

problem

Convert the relaxed LP to its dual

formStrong Duality

Theorem

Bilinear objective Linear constraints

Solve with global optimization solver

BARON


Compare the Optimal Functions of the Leaf Nodes

z1UB = z1

*UB + λ1UBθa + β1

UBθb

1

2

z2(2) = z2

*(2) + λ2(2)θa + β2

(2)θb

CR2(2)CR2

(1)

CR2(3)

CR1(1) CR1

UB

CR1UB CR2

(2) = CRint


Compare the Optimal Functions of the Leaf Nodes

min Єs.t. z1

UB = z2(2) + Є

z1UB = z1

*UB + λ1UBθa + β1

UBθb

z2(2) = z2

*(2) + λ2(2)θa + β2

(2)θb

θa,θb Є CRint

Redundancy test

on constraint z1UB ≥

z2(2)

Case 1: Problem is infeasible: z1UB is smaller in

CRintCase 2: Є > 0: the constraint is redundant. z2

(2) is smaller in CRint The optimal function is updated to be z2

(2) if node 2 is an integer node, otherwise, do not update.

Compare optimal function z1UB and z2

(2) in region CRint

Case 3: Є < 0: the constraint is not redundant. CRint is divided into two parts. z1

UB is smaller on one side and z2(2) is

smaller on the other side. The two regions are divided by z1

UB ≤ z2(2).


Advantage of Proposed Approach

The new mpLP approach can efficiently determine the

optimal functions and critical regions without retrieving the

LP tableaus and visiting the neighbor bases

Solve mpLP at only the leaf nodes in the B&B tree instead

of every node during the branch and bound procedure –

reduce the computational efforts significantly


Case Study: Single Uncertain Parameter

min z = 2x1 + 3x2 + 1.5x3 + 2x4 + 0.5x5

s.t. 2x1 + x2 + x3 ≥ 7 + Δa

2x2 + x4 + x5 ≥ 4

x3 + x4 – x5 ≥ 0

2x1 – x2 – x3 + x5 ≥ 4

1≤ x ≤ 3, xj Є (0,1), j = 3,4,5

Step 1:

3

2

x3 = 1 x3 = 0

x4 = 1 x4 = 0

0

12

11.5* 12(0,1,1)

Step 2: Linear sensitivity analysis on node 0 -- Δamax = 0

Step 3: For Δa = Δamax + ε, node 0 yields noninteger solution. Update the B&B tree.

Δa = 0


3

2

x3 = 1 x3 = 0

x4 = 1 x4 = 0

0

12

11.8*

12

λ = 3

x5 = 1 x5 = 0

12.1

3

2

x3 = 1 x3 = 0

1 0

012.2

12.1 12*

1 0

12.2

1 1 00

12*4

5

λ = 0

λ = 0

λ = 1

λ = 1 λ = 2

Step 2:

Δamax = min{Δabasis, min{ } }zp – z0

λp – λ0P

= min{ 2, min { }}0.32 ,

0.23 ,

0.23

0.23

=

Step 3:

For Δa = Δamax + ε, node 0 is intersected by node 2 & 3. Update the B&B tree

Step 3:

Step 2: ...

...



0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 211.5

12

12.5

13

13.5

14

(0,1,1)

(1,0,1) & (0,0,0)

(1,0,1)

(1,0,1)

Final optimal solution:

z*

Δa



Case Study: Multiple Uncertain Parameters

min z = -3x1 - 8x2 + 4y1 + 2y2

s.t. x1 + x2 ≤ 13 + θ1

5x1 - 4x2 ≤ 20

-8x1 + 22x2 ≤ 121 + θ2

4x1 + x2 ≥ 8

x1 – 10y1 ≤ 0

x2 – 15y3 ≤ 0

x ≥ 0, y Є (0,1), 0 ≤ θ1,θ2 ≤ 10Step 1:

1

y1 = 0 y1 = 1

y2 = 0 y2 = 1

2-8 -70.5*

(θ1, θ2) = (0, 0)


max (-13-θ1)d1 – 20d2 + (-121-θ2)d3 + 8d4 – d7 – d8 - zs.t. -d1 – 5d2 + 8d3 + 4d4 – d5 ≤ -3

-d1 + 4d2 – 22d3 + d4 – d6 ≤ -8

10d5 – d7 ≤ 4

15d6 – d8 ≤ 2

z ≥ -70.5 – 4.3333θ1 – 0.1667θ2

0 ≤ θ1, θ2 ≤ 10

Step 2: mpLP on the leaf nodes

Node 1z1

(1) = -70.5 – 4.3333θ1 – 0.1667θ2 at (θ1, θ2) = (0, 0)

obj = 16.74 (θ1, θ2) = (10, 0)

z1(2) = -97.0909 – 0.1667θ2 at (θ1, θ2) = (10, 0)



max (-13-θ1)d1 – 20d2 + (-121-θ2)d3 + 8d4 – d7 – d8 - zs.t. -d1 – 5d2 + 8d3 + 4d4 – d5 ≤ -3

-d1 + 4d2 – 22d3 + d4 – d6 ≤ -8

10d5 – d7 ≤ 4

15d6 – d8 ≤ 2

z ≥ -70.5 – 4.3333θ1 – 0.1667θ2

z ≥ -97.0909 – 0.3636θ2

0 ≤ θ1, θ2 ≤ 10obj = 0, stop

z1(1) ≥ z1

(2) 0.07333θ1 – 0.00333θ2 ≤ 0.45

The two critical regions are divided by



Node 2 z2 = 8

CR2 =θ1 ≤ 10

θ2 ≤ 10

Step 3: Compare critical regions and determine optimal functions

0.07333θ1 – 0.00333θ2 ≤ 0.45

θ2 ≤ 10

z1(1) = -70.5 – 4.3333θ1 –

0.1667θ2

CR1(1)

=

z1(2) = -97.0909 – 0.3636θ2

0.07333θ1 – 0.00333θ2 ≥ 0.45

θ1, θ2 ≤ 10CR1

(2) =

CR1(1) and CR2

CR1(1) n CR2 = CR1

(1)



min Єs.t. -97.0909 – 0.3636θ2 + Є = -8

0.07333θ1 – 0.00333θ2 ≥ 0.45

0 ≤ θ1, θ2 ≤ 10

Є > 0,z1

(2) ≤ z2 is redundant in CR1

(2)

min Єs.t. -70.5 – 4.3333θ1 - 0.1667θ2 + Є = -8

0.07333θ1 – 0.00333θ2 ≤ 0.45

θ2 ≤ 10

Є < 0, z1(1) ≤ z2

CR1(2) and CR2

CR1(2) n CR2 = CR1

(2)

CR1(1)

CR1(2)



0.07333θ1 – 0.00333θ2 ≤ 0.45

θ2 ≤ 10

z1(θ) = -70.5 – 4.3333θ1 – 0.1667θ2

CR1(1)

=

z2(θ) = -97.0909 – 0.3636θ2

0.07333θ1 – 0.00333θ2 ≥ 0.45

θ1, θ2 ≤ 10CR1

(2) =

y* = (1, 1)

Final optimal solution:



Uncertainty Analysis on the Objective Function Coefficients

minimize z = cxsubject to Ax θ x 0, xj Є (0, 1), j=1,…k

minimize z = (c + Δc)xsubject to Ax θx 0, xj Є (0, 1) , j=1,…k

z* = max{z(k) + λ(k)θa + β(k)θb}

Unlike the case of uncertain RHS, where the optimal objective value

Here, z* = min{z(k) + λ(k)c1 + β(k)c2}


maximize z – (c + Δc)x subject to Ax ≥ θ

z ≤ z(k) + λ(k)c1 + β(k)c2

c10 ≤ c1 ≤ c10 + Δc1

c20 ≤ c2 ≤ c20 + Δc2


maximize z – (c + Δc)x* subject to z = min{z(k) + λ(k)θa + β(k)θb}

c10 ≤ c1 ≤ c10 + Δc1

c20 ≤ c2 ≤ c20 + Δc2


where x* = argmin (c+Δc)xsubject to Ax ≥ θ

One level NLP


Uncertainty Analysis on the Constraint Coefficients

minimize z = cxsubject to Ax θ x 0, xj Є (0, 1), j=1,…k

minimize z = cxsubject to (A + ΔA)x θx 0, xj Є (0, 1) , j=1,…k

maximize cx* – zsubject to z = max{z(k) + λ(k)a1 + β(k)a2}

a10 ≤ a1 ≤ a10 + Δa1

a20 ≤ a2 ≤ a20 + Δa2


where x* = argmin cxsubject to Ax ≥ θ

At mpLP procedure, need to solve


Solve Bilevel Linear Programming Problem

The most popular method is “Kuhn-Tucker” approach

min F(x, y) = c1x + d1y

subject to A1x + B1y ≤ b1

min f(x, y) = c2x + d2y

subject to A2x + B2y ≤ b2

xЄX

yЄYReplace with its KKT condition and add to the upper level problem



Robust Optimization

MILP Sensitivity

Analysis

Expected Makespan/Profit

Model Robustness

Solution Robustness

Objective =


Robust Optimization

*Upper Partial Mean

S.Ahmed and N. Sahinidis. Robust process planning under uncertainty. 1998

stk(s,n) = stk(s,n-1) – dk(s,n) + P(s,i)bk(i,j,n-1) + cbk(i,j,n)

stk(s,n) stmax(s)

Vmin(i,j)wv(i,j,n) bk(i,j,n) Vmax(i,j)wv(i,j,n)

dk(s,n) + slackk(s) r(s)

Tfk(i,j,n) = Tsk(i,j,n) + (i,j)wv(i,j,n) + (i,j)bk(i,j,n)

Tsk(i,j,n+1) Tfk(i,j,n) – U(1-wv(i,j,n))

Tsk(i,j,n) Hk, Tfk(i,j,n) Hk

k Hk – PkHk, k 0k

n

(i,j)

Pkk

k

Pkslackk(s)k s

PkHk

k

minimize

Average MakespanModel Robustness

Solution Robustness

Unsatisfied Demand

subject to wv(i,j,n) 1


Multiobjective Optimization

Pareto Optimal Solution:

A point x*єC is said to be Pareto optimal if and only if there is no such xєC that fi(x) ≤ fi(x*) for all i={1,2,…,n} , with at least one strict inequality.

Min F(x) =

f1(x)f2(x)

fn(x)

::xєC

C = {x: h(x) = 0, g(x) ≤ 0, a ≤ x ≤ b}


Normal Boundary Intersection (NBI)

f1*

f2*

F*

Max t

g(x) ≤ 0h(x) = 0

a ≤ x ≤ b

x,t

Φω+ t n = F(x) – F*^s.t.

(Utopia point)

Min F(x) =f1(x)f2(x)

NBIω:

A point in the Convex Hull of Individual Minima (CHIM)

I. Das and J. Dennis. NBI: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems. 1996

Advantage: can produce a set of evenly distributed Pareto points independent of relative scales of the functions


Case Study

S1S1 S2S2 S3S3 S4S4

f1(x*) =6.4654

0.09

f2(x*) =11.5

00.66

f3(x*) =6.77500

mixingmixing reactionreaction purificationpurification

F* =6.46

00

stk(s,n) = stk(s,n-1) – dk(s,n) + P(s,i)bk(i,j,n-1) + cbk(i,j,n)

stk(s,n) stmax(s)

Vmin(i,j)wv(i,j,n) bk(i,j,n) Vmax(i,j)wv(i,j,n)

dk(s,n) + slackk(s) r(s)

Tfk(i,j,n) = Tsk(i,j,n) + (i,j)wv(i,j,n) + (i,j)bk(i,j,n)

Tsk(i,j,n+1) Tfk(i,j,n) – U(1-wv(i,j,n))

Tsk(i,j,n) Hk, Tfk(i,j,n) Hk

k Hk – PkHk, k 0

Pkkk

Pkslackk(s)k

s

PkHkk

minimize

(i,j)subject to wv(i,j,n) 1

Φ =

054

0.09

4.140

0.66

0.31500

054

0.09

4.140

0.66

0.31500

ω1

ω2

ω3 Pkk

k

Pkslackk(s)k s

PkHkk

- 6.46

=^+ t n

maximize t


6

8

10

12

0

20

40

600

0.2

0.4

0.6

0.8

1

(6.77, 50, 0)

(6.46, 54, 0.09)

(11.5, 0, 0.662)

Pareto Surface

Expected makespan

Satisfying demand

Rob

ustn

ess

Case Study

schedule 1 schedule 2 schedule 3


Case Study: Crude Oil Unloading

Crude Oil Marine Vessels

Storage

Tanks

ChargingTanks

CrudeDistillation

Units

OtherProduction

Units

ComponentStock Tanks

BlendHeader Finished

ProductTanks

Lifting/ShippingPoints

Problem 1: Crude-oil Unloading and Mixing

Problem 2: Production Stage

Problem 3: Product Blending and Distribution


Case Study: Crude Oil Unloading

60 65 70 75 80 85 900

0.5

1

1.5

2

2.5

3

3.5

Expected Cost

Expecte

d P

ositiv

e D

evia

tion B

A(87.23, 0.06)

(66.19, 2.14)


Acknowledgements

Financial SupportBOC Gases, NSF CAREER Award (CTS-9983406), Petroleum Research Fund, Office of Naval Research


Our Web Page:http://sol.rutgers.edu/staff/marianth

THANKS

Short-Term Scheduling under Uncertainty Marianthi Ierapetritou Department Chemical and Biochemical Engineering Piscataway, NJ 08854-8058.

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