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Short-Term Scheduling under Uncertainty

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Title: Short-Term Scheduling under Uncertainty


1
Short-Term Scheduling under Uncertainty
  • Marianthi Ierapetritou
  • Department Chemical and Biochemical Engineering
  • Piscataway, NJ 08854-8058

2
Process Operations Decision Making
Online Control
  • Objective
  • Identify and reduce bottlenecks at different
    levels
  • Integration of the whole decision-making process

Opportunity for Optimization
3
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

4
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

5
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

6
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
7
Increased Complexity Parameter Fluctuations
8
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
9
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
10
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

11
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

12
Uncertainty in Scheduling
13
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
14
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

15
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?

16
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
17
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

18
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
19
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
20
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)
21
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
22
Case Study 1
schedule 3
schedule 2 (optimal when d 50)
schedule 1 (optimal when d 50)
23
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)
24
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.
25
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
26
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
27
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

28
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

29
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
30
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
31
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
32
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)?
?
33
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
34
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
35
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
36
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).
37
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

38
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
39
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
40
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
41
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
42
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)
43
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
44
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
45
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,
46
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
47
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
48
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
49
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 ?
50
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
51
Preventive Scheduling
  • MILP Sensitivity Analysis
  • Robust Optimization

Expected Makespan/Profit
Model Robustness
Objective
Solution Robustness
52
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
53
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.
54
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
55
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
56
Case Study
(11.5, 0, 0.662)
Robustness
(6.46, 54, 0.09)
Satisfying demand
Expected makespan
(6.77, 50, 0)
Pareto Surface
57
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
58
Case Study Crude Oil Unloading
(66.19, 2.14)
(87.23, 0.06)
59
Acknowledgements
Financial Support BOC Gases, NSF CAREER Award
(CTS-9983406), Petroleum Research Fund, Office of
Naval Research
60
THANKS?
Our Web Page http//sol.rutgers.edu/staff/mariant
h
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