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Short-Term Scheduling under Uncertainty

- Marianthi Ierapetritou
- Department Chemical and Biochemical Engineering
- Piscataway, NJ 08854-8058

Process Operations Decision Making

Online Control

- Objective
- Identify and reduce bottlenecks at different

levels - Integration of the whole decision-making process

Opportunity for Optimization

Uncertain Parameters

- 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

Short-term Scheduling

- Process Plant Optimal Schedule
- Given Determine
- Raw Materials, Required Products, Task

Sequence, - Production Recipe, Unit Capacity Exact

Amounts of material - Processed
- Scheduling objectives
- Economic Maximize Profit, Minimize Operating

Costs, - Minimize Inventory Costs
- Time Based Minimize Makespan, Minimize Tardiness

Continuous Time Formulation

- 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

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)

??P?b(i,j,n-1) ??c?b(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,n1) ? 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

Objective Function

s

Allocation Constraints

(i,j)

Material Balances

Capacity Constraints

Demand Constraints

n

Duration Constraints

M.G.Ierapetritou and C.A.Floudas. Effective

continuous-time formulation for short-term

scheduling. 1. Multipurpose batch processes. 1998

Increased Complexity Parameter Fluctuations

Uncertainty in Short-Term Scheduling

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

Deterministic Schedule

55.56

separation

44.44

74.07

50.93

reaction 3

reaction 1

reaction 1

demand (product 2) 50

74.07

4.63

50.93

reaction 2

reaction 2

reaction 3

E(makespan) 8.15hr

50.00

heating

Standard Deviation 2.63

0

8

2

4

6

1

3

5

7

Robust Schedule

55.56

separation

64.60

10.03

50.93

reaction 3

reaction 1

reaction 2

demand (product 2) 50(1 60)

10.40

64.04

4.63

50.93

reaction 1

reaction 2

reaction 3

reaction 2

E(makespan) 7.24hr

50.00

heating

Standard Deviation 0.29

0

8

2

4

6

1

3

5

7

Literature Review Representative Publications

- Reactive Scheduling

Handles uncertainty by adjusting a schedule upon

realization of the uncertain parameters or

occurrence of unexpected events

- 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

- Stochastic Programming

Uncertainty 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

- Fuzzy Programming

Considers random parameters as fuzzy numbers and

the constraints are treated as fuzzy sets

- 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

- Robust Optimization

Produces robust solutions that are immune

against uncertainties

- 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

- MILP Sensitivity Analysis

Utilizes MILP sensitivity analysis methods to

investigate the effects of uncertain parameters

and provide a set of alternative schedules

- Z.Jia and M.G.Ierapetritou. Short-term

Scheduling under Uncertainty Using MILP

Sensitivity Analysis. Ind. Eng. Chem. Res. 2004,

43, 3782

Uncertainty in Scheduling

Preventive Scheduling

New alternative schedules

MILP sensitivity analysis framework

Data perturbation

LB/UB on objective function

Deterministic schedule

Robust optimization method

A set of solutions represent trade-off between

various objectives

model robustness

solution robustness

Preventive Scheduling

- MILP Sensitivity Analysis

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)

??P?b(i,j,n-1) ??c?b(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,n1) ? 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-integer Linear Programming

- Robust Optimization

Questions to Address

- What is the effect of processing time at the

objective value?

55

15

mixing

mixing

55

15

reaction

reaction

50

20

separation

separation

10

0

2

4

8

6

H (time horizon)

- Can the schedule accommodate the demand

fluctuation?

- How the capacity of the units affect the

production objective?

Inference-based MILP Sensitivity Analysis

minimize z cx subject to Ax ? a 0? x ? h,

xj integer, j1,k

minimize z (c ?c)x subject to (A ?A)x ?

a ?a 0? x ? h, xj integer, j1,k

Aim Determine under what condition z ? z - ?z

remains valid

Partial assignment at node p

Bound z ? z - ?z holds if there are s1P,,snP

that satisfy

- for the perturbations ?A and ?a

- for the perturbations ?c

?iP ?AijujP ?sj(uj uj) - ?i?ai ? rP sjP ?

?iP?Aij, sjP ? -qjP, j 1,,n rP -?qjPujP

?Pa zP ?zP

??cjujP - sjP(ujP ujP) ? -rP sjP ? -?cj, sjP ?

-qjP, j 1,,n qjP ?iPAij - ?iPcj

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

Proposed Uncertainty Analysis Approach

Solve the deterministic scheduling problem using

BB tree

MILP Sensitivity Analysis

Extract information from the leaf nodes

- Range of objective change for certain parameter

change

Robustness Estimation

Makespan minimization is considered as the

objective

Obtain sequence of tasks from original schedule

Generate random demands in expected range

Makespan to meet a particular demand is found

using the sequence of tasks derived from original

schedule Binary variables corresponding to

allocation of tasks are fixed Batch sizes and

Starting and Finishing times of tasks are allowed

to vary

Robustness under Infeasibility

Corrected Standard Deviation

Hact Hp if scenario is feasible

Hcorr if the scenario is infeasible

J.P.Vin and M.G.Ierapetritou. Robust short-term

scheduling of multiproduct batch plants under

demand uncertainty. 2001

Case Study 1

S1

S2

S3

S4

mixing

reaction

purification

Effect of demand d20, 100

-0.097 ?d ? ?H

dnom 50

Hnom 9.83h

H ? Hnom 0.097?d 12.73h

d 80

3.0

BB tree with nominal demand

wv(i1,j1,n0)

0

1

5.17

3.0

wv(i1,j1,n1)

0

0

1

1

7.65

5.83

7.16

infeasible

wv(i2,j2,n1)

0

0

0

1

1

1

8.14

7.16

7.16

10.16

8.33

infeasible

wv(i2,j2,n2)

0

0

0

0

1

1

1

1

9.87

8.83

8.83

9.98

9.83

9.83

infeasible

wv(i3,j3,n2)

infeasible

(Schedule 1)

Case Study 1

3.0

wv(i1,j1,n0)

0

1

5.17

3.0

wv(i1,j1,n1)

0

0

1

1

7.65

5.83

7.16

infeasible

wv(i2,j2,n1)

0

0

0

1

1

1

8.14

7.16

7.16

10.16

8.33

infeasible

wv(i2,j2,n2)

0

0

0

0

1

1

1

1

?

?

?

9.87

8.83

8.83

9.98

9.83

9.83

(12.13)

(12.73)

(17.97)

infeasible

infeasible

wv(i3,j3,n2)

(Schedule 1)

wv(i3,j3,n3)

schedule 2

schedule 3

schedule 1

Schedule Evaluation

10.77

10.91

Hnom(h)

9.83

11.56

Havg(h)

14.20

11.79

SDcorr

1.61

5.52

2.17

Case Study 1

schedule 3

schedule 2 (optimal when d 50)

schedule 1 (optimal when d 50)

Case Study 1

Effect of processing time T(i1,j1) 2.0, 4.0

Tnom 3.0

profitnom 71.52

profit ? profitnom 24.48?T 47.04

T 4.0

schedule 2

schedule 3

schedule 1

100

wv(i1,j1,n0)

65.27

65.27

profitnom

71.52

0

1

100

50

profitavg

66.98

65.17

64.61

wv(i1,j1,n1)

0

1

SDcorr

9.33

10.49

26.9

100

100

wv(i2,j2,n1)

0

0

1

1

100

96.05

50

50

wv(i2,j2,n2)

0

0

1

1

75

78.42

75

72.46

(75)

(75)

(62.11)

wv(i3,j3,n2)

0

1

50

78.42

wv(i3,j3,n3)

1

0

71.52

50

(Schedule 1)

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 dTy subject to Ax Dy ? b xL

? x ? xU ?L ? ? ? ?U x ? Rm, y ? (0,1)T

b b0 ?r

b?b0?Lr, b0?Ur

solved at b b0?Lr optimal solution (x,y)

Fix integer variables at y

LP sensitivity analysis

z(?) min cTx dTy subject to Ax Dy - ?r?

b cTx dTy z0 - ?? 0 ?yi - ?yi ? ?F1? - 1 xL

? x ? xU ?L ? ? ? ?U x ? Rm, y ? (0,1)T

z(?) z0 ?? ?L ? ?L ? ? ? ?U ? ?U

Integer cut to exclude current optimal solution

i?F1

i?F 0

break point ?, new optimal solution (x, y)

A.Pertsinidis et al. Parametric optimization of

MILP programs and a framework for the parametric

optimization of MINLPs. 1998

Multiparametric MILP

Solve the fully relaxed problem

Select a branching variable

Based on simplex algorithm, check the neighboring

bases of the LP tableau

Solve the mpLP at the nodes

Compare the solution with the current UB, update

the optimal function in the uncertain space

? J. Acevedo and E.N.Pistikopoulos. A

Multiparametric Programming Approach for Linear

Process Engineering Problems under Uncertainty.

1997

Shortcomings of Existing Approach

- Solve mpLP at every node in the BB tree during

the branch and bound procedure can be a

computationally expensive effort

- mpLP approach requires retrieving the LP

tableaus and visiting the neighbor bases

Proposed Analysis on the RHS for MILPs

minimize z cx subject to Ax ? a x ? 0, xj

? (0, 1), j1,k

minimize z cx subject to Ax ? a ?a x ?

0, xj ? (0, 1) , j1,k

Develop a framework to investigate the effect of

?a on the optimal solution x and objective value z

- A set of optimal integer solutions
- Critical regions
- Optimal functions

Proposed Approach Single Uncertain Parameter

Solve the original problem at the nominal value

using a branch and bound method

Find ?amax that leaves the structure of the tree

unchanged

Collect zp, ?p at each leaf node p

For a ?amax e update the BB tree

Determine ?amax and Update the BB Tree

where node 0 is the optimal node

?amax min?abasis, min

P

Update the BB tree at a ?amax e

0

The new optimal node can be

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

Proposed Approach Multiple Uncertain Parameters

Solve the original problem at the nominal value

using a branch and bound method

mpLP at the leaf nodes

mpLP algorithm

Compare the critical regions with the

current upper bounds Update the BB tree

mpLP Algorithm at the Leaf Nodes

At each iteration, solve

maximize cx z subject to z maxz(k)

?(k)?a ß(k)?b a0 ?a a0 ?a

b0 ?b b0 ?b

Bilevel Linear Programming

where x argmin cx subject to Ax ?

cx

cx

current optimal functions

maxz(k) ?(k)?

?

mpLP Algorithm at the Leaf Nodes

check if there is any point at which maxz(k)

?(k)?a ß(k)?b is less than cx

maximize min cx Ax ? ? z subject to z ?

z(k) ?(k)?a ß(k)?b a0 ?a a0

?a b0 ?b b0 ?b

current optimal functions

Include an additional constraint z ? z(k1)

?(k1)?a ß(k1)?b to above problem

Stop when the objective 0

Solve the Bilevel Programming Problem

Convert the relaxed LP to its dual form

max ?y s.t. ATy c a0 ?a a0

?a b0 ?b b0 ?b Y ? 0

min cx s.t. Ax ? ? a0 ?a a0

?a b0 ?b b0 ?b x ? 0

Strong Duality Theorem

Replace the inside optimization problem by its

dual

Solve with global optimization solver BARON

Bilinear objective Linear constraints

max ?y - z s.t. ATy c z ? z(k)

?(k)?a ß(k)?b a0 ?a a0 ?a

b0 ?b b0 ?b Y ? 0

Single optimization problem

Compare the Optimal Functions of the Leaf Nodes

1

2

CR2(1)

CR2(2)

CR1(1)

CR1UB

CR2(3)

z1UB z1UB ?1UB?a ß1UB?b

z2(2) z2(2) ?2(2)?a ß2(2)?b

CR1UB CR2(2) CRint

Compare the Optimal Functions of the Leaf Nodes

Compare optimal function z1UB and z2(2) in region

CRint

min ? s.t. z1UB z2(2) ? z1UB z1UB

?1UB?a ß1UB?b z2(2) z2(2) ?2(2)?a

ß2(2)?b ?a,?b ? CRint

Redundancy test on constraint z1UB z2(2)

Case 1 Problem is infeasible z1UB is smaller

in CRint

Case 2 ? gt 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.

Case 3 ? lt 0 the constraint is not redundant.

CRint is divided into two parts. z1UB is smaller

on one side and z2(2) is smaller on the other

side. The two regions are divided by z1UB z2(2).

Advantage of Proposed Approach

- Solve mpLP at only the leaf nodes in the BB

tree instead of every node during the branch and

bound procedure reduce the computational

efforts significantly

- The new mpLP approach can efficiently determine

the optimal functions and critical regions

without retrieving the LP tableaus and visiting

the neighbor bases

Case Study Single Uncertain Parameter

min z 2x1 3x2 1.5x3 2x4 0.5x5 s.t.

2x1 x2 x3 7 ?a 2x2 x4 x5 4 x3

x4 x5 0 2x1 x2 x3 x5 4 1 x

3, xj ? (0,1), j 3,4,5

Step 1

?a 0

x3 1

x3 0

2

12

x4 1

x4 0

3

0

11.5

12

(0,1,1)

Step 2

Linear sensitivity analysis on node 0 -- ?amax 0

For ?a ?amax e, node 0 yields noninteger

solution. Update the BB tree.

Step 3

Case Study Single Uncertain Parameter

Step 2

x3 1

x3 0

2

?amax min?abasis, min

12

x4 1

x4 0

? 0

P

min 2, min

3

0

12

x5 1

x5 0

? 0

11.8

12.1

? 3

? 1

Step 3

For ?a ?amax e, node 0 is intersected by node

2 3. Update the BB tree

x3 1

x3 0

2

1

0

1

0

...

3

0

12.2

12

4

Step 2

? 1

? 2

1

1

0

0

...

Step 3

5

12.1

12

12.2

Case Study Single Uncertain Parameter

Final optimal solution

14

13.5

(1,0,1)

13

z

12.5

(1,0,1)

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

12

(0,1,1)

11.5

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2

?a

Case Study Multiple Uncertain Parameters

min z -3x1 - 8x2 4y1 2y2 s.t. x1

x2 13 ?1 5x1 - 4x2 20 -8x1 22x2 121

?2 4x1 x2 8 x1 10y1 0 x2 15y3

0 x 0, y ? (0,1), 0 ?1,?2 10

Step 1

(?1, ?2) (0, 0)

y1 0

y1 1

y2 0

y2 1

1

2

-8

-70.5

Case Study Multiple Uncertain Parameters

mpLP on the leaf nodes

Step 2

z1(1) -70.5 4.3333?1 0.1667?2 at (?1,

?2) (0, 0)

Node 1

max (-13-?1)d1 20d2 (-121-?2)d3 8d4 d7

d8 - z s.t. -d1 5d2 8d3 4d4 d5

-3 -d1 4d2 22d3 d4 d6 -8 10d5 d7

4 15d6 d8 2 z -70.5 4.3333?1

0.1667?2 0 ?1, ?2 10

obj 16.74 (?1, ?2) (10, 0)

z1(2) -97.0909 0.1667?2 at (?1, ?2)

(10, 0)

Case Study Multiple Uncertain Parameters

max (-13-?1)d1 20d2 (-121-?2)d3 8d4 d7

d8 - z s.t. -d1 5d2 8d3 4d4 d5

-3 -d1 4d2 22d3 d4 d6 -8 10d5 d7

4 15d6 d8 2 z -70.5 4.3333?1

0.1667?2 z -97.0909 0.3636?2 0

?1, ?2 10

obj 0, stop

The two critical regions are divided by

z1(1) z1(2) 0.07333?1 0.00333?2

0.45

Case Study Multiple Uncertain Parameters

z1(2) -97.0909 0.3636?2

z1(1) -70.5 4.3333?1 0.1667?2

0.07333?1 0.00333?2 0.45

0.07333?1 0.00333?2 0.45

CR1(1)

CR1(2)

?2 10

?1, ?2 10

z2 8

Node 2

?1 10

CR2

?2 10

Step 3

Compare critical regions and determine optimal

functions

CR1(1) n CR2 CR1(1)

CR1(1) and CR2

Case Study Multiple Uncertain Parameters

min ? s.t. -70.5 4.3333?1 - 0.1667?2

? -8 0.07333?1 0.00333?2 0.45

?2 10

CR1(1)

z1(1) z2

? lt 0,

CR1(2) n CR2 CR1(2)

CR1(2) and CR2

min ? s.t. -97.0909 0.3636?2 ? -8

0.07333?1 0.00333?2 0.45 0 ?1, ?2

10

CR1(2)

z1(2) z2 is redundant in CR1(2)

? gt 0,

Case Study Multiple Uncertain Parameters

Final optimal solution

y (1, 1)

z1(?) -70.5 4.3333?1 0.1667?2

0.07333?1 0.00333?2 0.45

CR1(1)

?2 10

z2(?) -97.0909 0.3636?2

0.07333?1 0.00333?2 0.45

CR1(2)

?1, ?2 10

Uncertainty Analysis on the Objective Function

Coefficients

minimize z cx subject to Ax ? ? x ? 0, xj

? (0, 1), j1,k

minimize z (c ?c)x subject to Ax ? ? x

? 0, xj ? (0, 1) , j1,k

Unlike the case of uncertain RHS, where the

optimal objective value

z maxz(k) ?(k)?a ß(k)?b

Here, z minz(k) ?(k)c1 ß(k)c2

mpLP Algorithm at the Leaf Nodes

maximize z (c ?c)x subject to z

minz(k) ?(k)?a ß(k)?b c10 c1

c10 ?c1 c20 c2 c20 ?c2

Bilevel Linear Programming

where x argmin (c?c)x subject to Ax ?

maximize z (c ?c)x subject to Ax ?

z z(k) ?(k)c1 ß(k)c2 c10

c1 c10 ?c1 c20 c2 c20 ?c2

One level NLP

Uncertainty Analysis on the Constraint

Coefficients

minimize z cx subject to Ax ? ? x ? 0, xj

? (0, 1), j1,k

minimize z cx subject to (A ?A)x ? ? x

? 0, xj ? (0, 1) , j1,k

At mpLP procedure, need to solve

maximize cx z subject to z maxz(k)

?(k)a1 ß(k)a2 a10 a1 a10

?a1 a20 a2 a20 ?a2

Bilevel Linear Programming

where x argmin cx subject to Ax ?

Solve Bilevel Linear Programming Problem

The most popular method is Kuhn-Tucker approach

min F(x, y) c1x d1y subject to A1x B1y

b1

x?X

min f(x, y) c2x d2y subject to A2x B2y

b2

Replace with its KKT condition and add to the

upper level problem

y?Y

Preventive Scheduling

- MILP Sensitivity Analysis

- Robust Optimization

Expected Makespan/Profit

Model Robustness

Objective

Solution Robustness

Robust Optimization

Average Makespan

Model Robustness

minimize

Solution Robustness

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

stk(s,n) stk(s,n-1) dk(s,n)

??P(s,i)?bk(i,j,n-1) ??c?bk(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,n1) ? 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

(i,j)

Unsatisfied Demand

n

Upper Partial Mean

k

? S.Ahmed and N. Sahinidis. Robust process

planning under uncertainty. 1998

Multiobjective Optimization

f1(x)

f2(x)

Min F(x)

x?C

fn(x)

C x h(x) 0, g(x) 0, a x b

Pareto Optimal Solution

A point x?C is said to be Pareto optimal if and

only if there is no such x?C that fi(x) fi(x)

for all i1,2,,n , with at least one strict

inequality.

Normal Boundary Intersection (NBI)

f1(x)

Advantage can produce a set of evenly

distributed Pareto points independent of relative

scales of the functions

Min F(x)

f2(x)

NBI?

Max t

x,t

F? t n F(x) F

s.t.

h(x) 0

f2

g(x) 0

a x b

f1

F

A point in the Convex Hull of Individual Minima

(CHIM)

(Utopia point)

? I. Das and J. Dennis. NBI A new method for

generating the Pareto surface in nonlinear

multicriteria optimization problems. 1996

Case Study

S1

S2

S3

S4

mixing

reaction

purification

6.46 54 0.09

11.5 0 0.66

6.77 50 0

f1(x)

f3(x)

f2(x)

6.46 0 0

0 54 0.09

4.14 0 0.66

0.31 50 0

F

F

?PkHk

- 6.46

?1 ?2 ?3

0 54 0.09

4.14 0 0.66

0.31 50 0

k

?Pk?slackk(s)

t n

k

s

?Pk?k

k

Case Study

(11.5, 0, 0.662)

Robustness

(6.46, 54, 0.09)

Satisfying demand

Expected makespan

(6.77, 50, 0)

Pareto Surface

Case Study Crude Oil Unloading

Problem 3 Product Blending and Distribution

Problem 1 Crude-oil Unloading and Mixing

Problem 2 Production Stage

Crude Oil Marine Vessels

Storage Tanks

Charging Tanks

Crude Distillation Units

Other Production Units

Component Stock Tanks

Blend Header

Finished Product Tanks

Lifting/Shipping Points

Case Study Crude Oil Unloading

(66.19, 2.14)

(87.23, 0.06)

Acknowledgements

Financial Support BOC Gases, NSF CAREER Award

(CTS-9983406), Petroleum Research Fund, Office of

Naval Research

THANKS?

Our Web Page http//sol.rutgers.edu/staff/mariant

h