COMPUTER-AIDED DESIGN,
ENGINEERING, AND MANUFACTURING
Systems Techniques And Applications
THE
DESIGN
oF
MANUFACTURING
SYSTEMS
VOLUME
V
© 2001 by CRC Press LLC
COMPUTER-AIDED DESIGN,
ENGINEERING, AND MANUFACTURING
Systems Techniques And Applications
VOLUME
Editor
CORNELIUS LEONDES
Boca Raton London New York Washington, D.C.
CRC Press
THE DESIGN OF
MANUFACTURING
SYSTEMS
V

This book contains information obtained from authentic and highly regarded sources. Reprinted material is
quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts
have been made to publish reliable data and information, but the author and the publisher cannot assume
responsibility for the validity of all materials or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or
mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval
system, without prior permission in writing from the publisher.

any event, the great breadth of the field certainly suggests the requirement for seven distinctly titled and
well-integrated volumes for an adequately comprehensive treatment. The seven volume titles are:
1. Systems Techniques and Computational Methods
2. Computer-Integrated Manufacturing
3. Operational Methods in Computer-Aided Design
4. Optimization Methods for Manufacturing
5. The Design of Manufacturing Systems
6. Manufacturing Systems Processes
7. Artificial Intelligence and Robotics in Manufacturing
The contributors to this volume clearly reveal the effectiveness and great significance of the techniques
available and, with further development, the essential role that they will play in the future. I hope that
practitioners, research workers, students, computer scientists, and others on the international scene will
find this set of volumes to be a unique and significant reference source for years to come.

Cornelius T. Leondes
Editor

© 2001 by CRC Press LLC

Editor

Cornelius T. Leondes

, B.S., M.S., Ph.D., is an Emeritus Professor at the School of Engineering and Applied
Science, University of California, Los Angeles. Dr. Leondes has served as a member or consultant on
numerous national technical and scientific advisory boards. He has served as a consultant for numerous
Fortune 500 companies and international corporations, published over 200 technical journal articles,
and edited and/or co-authored over 120 books. Dr. Leondes is a Guggenheim Fellow, Fulbright Research
Scholar, and Fellow of IEEE. He is a recipient of the IEEE Baker Prize, as well as its Barry Carlton Award.



University of Illinois at Urbana-
Champaign
Urbana, Illinois

Necdet Geren

University of Çukurova
Adana, Turkey

Klaus Henning

University of Technology (RWTH)
Aachen, Germany

Bao Sheng Hu

Xi’an Jiaotong University
Xi’an, China

Mark A. Lawley

Purdue University
West Lafayette, Indiana

T. Warren Liao

Louisiana State University
Baton Rouge, Louisiana



Contents

Preface

Chapter 1

Long-Range Planning of Chemical
Manufacturing Systems

Shabbir Ahmed and Nikolaos V. Sahinidis

Chapter 2

Feature-Based Design in Integrated Manufacturing

Venkat Allada

Chapter 3

Flexible Factory Layouts: Issues in Design, Modeling,
and Analysis

Saifallah Benjaafar

Chapter 4

Structural Control of Large-Scale Flexibly Automated
Manufacturing Systems


Petri Net Modeling in Flexible Manufacturing Systems
with Shared Resources

Ke Yi Xing and Bao Sheng Hu
Z
ˇ
zˇ

© 2001 by CRC Press LLC

1

Long-Range Planning
of Chemical
Manufacturing

Systems

1.1 Introduction

1.2 The Long-Range Planning Problem

General Formulation

1.3 Deterministic Models

An MILP Model • Extensions of the MILP Model

1.4 Hedging against Uncertainty


Nikolaos V. Sahinidis

1

University of Illinois
at Urbana-Champaign
© 2001 by CRC Press LLC

1.2 The Long-Range Planning Problem

Let us consider a plant comprising several processes to produce a set of chemicals for sale. Each process
intakes a number of raw materials and produces a main product along with some by-products. Any of
these main or by-products could be the raw materials for another process. We, thus, have a list of chemicals
consisting of the main products or by-products that we wish to sell as well as ingredients necessary for
the production of each chemical. We might then contemplate the in-house production of some of the
required ingredients, forcing us to consider another tier of ingredients and by-products. The listing
continues until we have considered all processes which may relate to the ultimate production of the
products initially proposed for sale. At this point, the final list of chemicals will contain all raw materials
we consider purchasing from the market, all products we consider offering for sale on the market, and
all possible intermediates. The plant can then be represented as a network comprised of nodes repre-
senting processes and the chemicals in the list, interconnected by arcs representing the different alterna-
tives that are possible for processing, and purchases to and sales from different markets.
The process planning problem then consists of choosing among the various alternatives in such way
as to maximize profit. Once we know the prices of chemicals in the various markets and the operating
costs of processes, the problem is then to decide the operating level of each process and amount of each
chemical in the list to be purchased and sold to the various markets. The problem in itself grows
combinatorially with the number of chemicals and processes and is further complicated once we start
planning over multiple time periods.
Let us now consider the operation of the plant over a number of time periods. It is reasonable to
expect that prices and demands of chemicals in various markets would fluctuate over the planning

ϭ
1,

NP

).

j

The set of

NC

chemicals that interconnect the processes (

j

ϭ
1,

NC

).


ϭ
1,

NT

).

Variables

E

it

Units of expansion of process

i

at the beginning of period

t

.

P

jlt


S

jlt

Units of chemical

j

sold to market

l

at the end of period

t

.

W

it

Operating level of process

i

in period

t



W

it

) The cost model for the operation of process

i

over period

t

as a function of the operating
level.

SALE

jlt

(

S

jlt

) The sales price model for chemical

j



t

as a function of the
purchase quantity.
The mass balance model for the output chemical

j

from process

i

as a function of the
operating level.
The mass balance model for the input chemical

j

for process

i

as a function of the
operating level.

Parameters

Lower and upper bounds for the availability (purchase amount) of chemical


(1.1)
subject to
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
(1.7)
␺
ij
O
W
it
()
␺
ij
I
W
it
()
a
jlt
L
a
jlt
U
,
d
jlt
L

P
jlt
()Ϫ[]
lϭ1
NM
Α
jϭ1
NC
Α



ϩ
Q
it
Q
itϪ1
E
it
ϩϭ i 1NP t 1 NT,ϭϭ
W
it
Q
it
Յ i 1NP t 1 NT,ϭϭ
P
jlt
lϭ1
NM
Α

jlt
a
jlt
U
ՅՅ j 1, NC; l 1NM t 1 NT,ϭϭϭ
a
jlt
L
S
jlt
a
jlt
U
ՅՅ j 1, NC; l 1 NM t 1 NT,ϭϭϭ
E
it
Q
it
W
it
,, 0Ն i 1NP t 1 NT,ϭϭ
© 2001 by CRC Press LLC
The objective function is to maximize the difference between the sales revenues of the final products
and the investment, operating, and raw material costs. Constraint (1.2) in the preceding formulation
defines the total capacity available at period t as a sum of capacity available in period t Ϫ 1 and the
capacity expansion at the beginning of period t. The parameter Q
i0
represents the initial capacity, that is,
at t ϭ 0. The condition that the operating level of any process cannot exceed the installed capacity is
modeled by constraint (1.3). Eq. (1.4) expresses mass balances for chemicals across processes and markets.

2. Forecasts for prices and demands of chemicals, as well as investment and operating costs over the
planning horizon, are given.
3. The planning horizon consists of a finite number of time periods.
4. Linear models are assumed for mass balances across processes. The functions and
in Eq. (1.4) are replaced by linear proportionality constants.
5. Fixed charge cost models are used to model investment costs. These models assume a fixed charge
associated with installation of new capacity and a variable charge proportional to the capacity
P
jlt
lϭ1
NM
Α
␺
ij
O
W
it
()
iϭ1
NP
Α
I
jtϪ1
ϩϩ
S
jlt
lϭ1
NM
Α
ϭ

), OPER
it
(W
it
), SALE
jlt
(S
jlt
), and PURC
jlt
(P
jlt
) in the objective (1.1) are replaced
by appropriate linear functions.
6. No inventories are carried over time periods because the length of each period is assumed to be
rather long.
Supplementary Notation
The notation used here is the same as in the section on general formulation with the inclusion of the
following parameters and variables.
Parameters
Forecasted buying and selling prices of chemical j in market l in period t.
Input and output proportionality constants for chemical j in process i.
␣
it
Per unit expansion cost for process i at the beginning of period t.
␤
it
Fixed cost of establishing or expanding process i at the beginning of period t.
␦
it

jlt
␥
jlt
,
␮
ij
I
␮
ij
O
,
E
it
L
E
it
U
,
max NPV
␣
it
E
it
␤
it
y
it
␦
it
W



P
jlt
lϭ1
NM
Α
␮
ij
O
W
it
tϭ1
NP
Α
ϩ
S
jlt
lϭ1
NM
Α
ϭ
␮
ij
I
W
it
tϭ1
NP
Α

The case of shut down of an existing plant results when the variable W
it
takes a value of zero after a
given time period t. The economics of shut down can be modeled by the inclusion of the following
variables and constraints.
Variables
⑀
it
A 0–1 integer variable. If process i is to be shut down at the beginning of time period t, then
⑀
it
ϭ 1, else
⑀
it
ϭ 0.
CP
it
Available plant capacity of process i at time of shut down t. If process i is not shut down at time
t, then CP
it
ϭ 0.
Constraints
(1.15)
(1.16)
(1.17)
(1.18)
(1.19)
(1.20)
where ϭ . When process i has been decided to be shut down at the beginning of period t,
that is,

iϭ1
NP
Α
CI t()Յ tTЈ 12,… NT,,{}ʚʦ
␣
it
␤
it
CP
it
Q
itϪ1
Յ i 1 NP t, 1 NT 1ϩ,ϭϭ
CP
it
⑀
it
U
it
Յ i 1 NP t, 1 NT 1ϩ,ϭϭ
Q
it
CP
it
1
⑀
it
Ϫ()U
itϪ1
ϩՅ i 1 NP t, 1 NT 1ϩ,ϭϭ

it
© 2001 by CRC Press LLC
Solution Strategies
Model P presented in the previous section is a mixed integer linear program and typically can be solved
by a branch and bound algorithm using linear programming (LP) relaxations. This is the basic algorithm
employed by most commercial codes (e.g., LINDO, ZOOM, OSL, CPLEX, SCICONIC) for these types
of problems. The branch and bound algorithm for mixed integer linear programs with integer variables
restricted to binary values is described next.
The Branch and Bound Algorithm
The main idea for the branch and bound algorithm for solving model P is that of divide and conquer. The
original problem is branched into two subproblems by selecting one of the integer variables and restricting
it to 0 and 1 in the two subproblems, respectively. The resulting subproblems can be further subdivided
by selecting another branching variable. In this way, the original problem is partitioned in the form of
a binary tree, where the root node denotes the original problem and each subsequent node represents
an easier subproblem. Because the binary variables can assume only 0–1 values and the nodes of the tree
represent all possible combinations of these values, the optimal solution of P, if it exists, must be in one
of the nodes. Note, however, that this tree can have have nodes, where NP ϫ NT is the number
of binary variables (y
it
). To avoid complete enumeration of the tree, we have to determine whether a
given node is required to be partitioned further. We can then prune the nodes that need not be further
examined, thus reducing computational effort. The process is called fathoming and is achieved by solving
an LP relaxation of the MILP subproblem in this node. The LP relaxation is obtained by replacing each
integrality constraint y
it
෈ {0, 1} in the MILP by the continuous constraints 0 Յ y
it
Յ 1 ∀i,t. If the
relaxed subproblem is infeasible, this indicates that the MILP in this node is also infeasible. If the relaxed
solution is no better than the best previously obtained integer solution, we infer that the current node

a. If Յ , then set S
kϩ1
ϭ S
k
\P
k
.
b. If Ͼ and ∀i,t, then let S
k+1
ϭ S
k
\P
k
and ϭ
c. If Ͼ and ∀i,t, then select a branching variable where (i
k
, t
k
)
ʦ and construct the following subproblems from problem P
k
:
Set S
kϩ1
ϭ .
Set k ← k ϩ 1 and GOTO 1.
The preceding method might be computationally expensive for the planning model because, for realistic
problems, the number of variables is large. For example, a network with 40 processes, 50 chemicals, 2 markets,
and 5 time periods would involve 200 binary variables and approximately 1000 continuous variables and
2

k
it: y
it
k
01,{},{}
P
k
0
P
k
ϭ y
i
k
t
k
0ϭ{}ʝ
P
k
1
P
k
y
i
k
t
k
1ϭ{}ʝϭ
S
k
\P

sponding capacities set to the minimum required in order to serve demand during all subsequent time
periods.
Additional computational gains can be obtained by including constraint (1.13), where NEXP(i) can
be obtained by solving the following MILP for each i
(1.22)
subject to
where U is a large positive quantity.
The preceding problem calculates the maximum number of expansions whose cost is less than or equal
to the maximum cost (in the worst-case sense) of any given expansion. The first constraint implies that
the cost of the expansions cannot exceed the investment cost of process i at maximum capacity Q
imax
with the ‘‘worst’’ coefficients,
␣
imax
and
␤
imax
. Owing to the discount factors, these coefficients usually corre-
spond to period 1. Q
imax
is the minimum capacity required to serve maximum possible demand. This
demand can be found by maximizing the operating level subject to the material balance constraints. From
the solution of the preceding small-scale MILPs, constraint (1.13) can be added to the general model.
Both constraints (1.13) and (1.21) help in reducing the gap between the relaxed LP and MILP solutions
so as to decrease the computational effort of the branch and bound method. However, for large-scale
problems, these provisions may not be sufficient. Furthermore, when Q
imax
exceeds , the problem in
(1.22) will often underestimate the maximum number of expansions.
Strong Cutting Planes

imax
Յ
␤
imax
ϩ
E
it
tϭ1
NT
Α
Q
imax
ϭ
0 E
it
Uy
it
ՅՅ t 1 NT,ϭ
y
it
01,{}ʦ t 1 NT,ϭ
E
i1
U
© 2001 by CRC Press LLC
additional valid inequalities which attempt to chop off the solution point from the space of the LP
relaxation polyhedron. The process is repeated, thereby reducing the gap between the MILP and its
LP relaxation.
A network substructure in the model P can be identified by making the following substitution.
With these, we obtain the following substructure for each process in each time period

E
it
L
␤
it
ϩϭ
u
it
␣
it
E
it
U
␤
it
ϩϭ





i ϭ 1 NP, t ϭ 1 NT,
S
it
xy,(): x
it
iϭ1
NP
Α
CI t()l

Α
CI t()
Յ
x
it
u
it
␭
y
Ϫ()
+
1 y
it
Ϫ()ϩ[]
iC
t
ʦ
Α
x
it
u
it
␭
it
Ϫ()
+
1 y
it
Ϫ()ϩ[]
iR

u
it
{}and u
t
␭
t
0ϾϾϭ
x
ء
y
ء
,()
max ␨
t
1 y
it
ء
Ϫ()z
i
Ϫ{}
iϭ1
NP
Α
ϭ
© 2001 by CRC Press LLC
subject to
where z
i
ϭ 1 if i ʦ C
t

problem depends on the partitioning of the variables. The natural choice for this partitioning is
1. Complicating variables for the master problem: u ϭ [y
it
].
2. Remaining variables for the LP subproblems: v ϭ [E
it
, P
jlt
, Q
it
, S
jlt
, W
it
].
However, with this variable partition, the master problem is often too relaxed. In order to strengthen the
bounds predicted by the master problem, the variable partitioning can be redefined as follows:
1. Complicating variables for the master problem: u ϭ [y
it
, Q
it
, E
it
].
2. Remaining variables for the LP subproblems: v ϭ [P
jlt
, S
jlt
, W
it

it
ء
0Ն
u
1
NPV
U
NPV
L
u
k
NPV
k
v
k
V
L
NPV
L
NPV
k
u
kϩ1
NPV
U
max
u,
␮
␮
ϭ

appeared to be a combination of integer cuts, strong cutting plane generation, and branch and bound.
Reformulation to Exploit Lot Sizing Substructures
Aiming at improving the LP bounds, Sahinidis and Grossmann [13] present a reformulation of the
general model P by exploiting the lot sizing substructures. To see this substructure, let us fix all the
chemical flows (W
it
, P
jlt
, S
jlt
) in the network in such a way that the mass balance constraint (1.10) is
satisfied for all time periods. Then, every process can be isolated from the rest of the network and the
planning problem for each process i becomes: “Find the cheapest capacity expansion sequence for process
i (E
it
, t ϭ 1, NT) which will allow the production of the specified operating level and flow of chemicals
(W
it
,P
jlt
,S
jlt
).” The equivalent lot sizing reformulation to the preceding problem becomes apparent with
the following substitutions.
Let
(1.26)
(1.27)
where W
i0
ϭ Q

r
NPV
L
NPV
U
SQ
it
Q
it
W
it
i 1 NP t,ϭ 1 NT,ϭϪϭ
d
it
{W
it
max
Tϭ0 t Ϫ1,
Ϫ W
iT
}
ϩ
i 1 NP t,ϭ 1 NT,ϭϭ
min

␣
it
E
it
␤

it
01,{} t 1 NT,ϭʦ
© 2001 by CRC Press LLC
in period t to serve production requirements up to period
␶
(
␶
Ն t). Model LSR-i then becomes:
Model MLSR-i
subject to
(1.28)
(1.29)
where C
it
␶
are upper bounds for the required capacity expansions and can be obtained by
The evaluation of d
it
by (1.27) results in nonconvexities in the model. The alternative suggested in
reference [13] is to a priori estimate the bounds C
it
␶

as follows. Constraint (1.29) can be relaxed as
Furthermore, C
it
␶
in constraint (1.28) can be estimated by which is given by
where ⍀
it

t 1 NT
␶
tՆ,ϭՆ
␾
itr
C
it
␶
y
it
t 1 NT
␶
, tՆϭՅ
␾
i1t
␶ϭ1
t
Α

Ն
C
ilt
t 1 NT,ϭ
E
it
␾
it
␶
, 0 t 1 NT
␶

it
␶
,
C
ˆ
it
␶
min E
it
U
, max ⍀
iT
{}Q
i0
max
␻
iT
Q
i0
Ϫ{}
ϩ
ϩ()Ϫ[]
ϩ
ϭ
Tϭt,
␶
Tϭ1 tϪ1,
max NPV as given in (1.9)
E
it

it
␶
, find one or more constraints from among (1.30) and (1.31) that are violated
by the solution to the current LP relaxation. If such inequalities are found, append them to the
current LP and go to Step 2. Else, go to Step 3.
2 Solve the new LP relaxation. If NPVЈ Ϫ NPV Ͼ
⑀
(where NPV is the current LP solution and
⑀
is a prescribed tolerance), then set NPVЈϭ NPV and repeat Steps 1 and 2. Otherwise, go to Step 3.
3 Start a branch and bound procedure or any other exact algorithm to find the optimum to the
current LP formulation.
In the preceding algorithm, the violated constraints are added on an as needed basis. Because the
number of
␾
variables is quite high, a large number of constraints may need to be added in each step.
Projection Approach
The increased number of variables in RP can be reduced by projecting the problem from the (E,W,P,S,y,
␾
)
space onto the (E,W,P,S,y) space. Let x ϭ

(E,W,P,S,y) and P be the polyhedron defining the feasible space
of RP: P ϭ {(x,
␾
) : Ax ϩ G
␾
Յ b,
␾
Ն 0, x ʦ X}. Also, let C

0 i 1 NP t, 1 NT
␶
,ϭϭ tՆՆ
NT
2
v
1
v
2
v
H
v
h
v
h
max NPV as given in (1.9)
W
it
␴
i
␶
t
E
i
␶
1
␴
i
␶
t

constraints in PP increases exponentially with the number of time periods. Computational results in [6]
using branch and bound, indicate that for up to four time periods, RP and PP require smaller CPU times
and fewer search nodes than P. However, for more than four time periods, P performs best and PP worst
in terms of CPU times. This happens in spite of the fact that RP and PP require fewer search nodes than
P. The increased CPU time for RP and PP can be attributed to the large number of additional constraints
involved. However, when a constraint generation scheme is coupled with branch and bound for solving
RP and PP, formulation RP outperforms P whereas PP outperforms both P and RP as the problem size
increases. The success of the projection cutting plane approach is owing to the fact that only a small
fraction of the projection constraints (1.33) is sufficient to significantly reduce the relaxation gap.
Remarks
Previously we discussed some of the solution strategies for model P. Bounding and integer cuts, strong
cutting planes, and Benders decomposition were described as methods of direct solution of P. Two
nonconventional reformulations of the same model, RP and PP, and their solution strategies were also
discussed. In summary, we can make the following remarks.
1. Straightforward branch and bound strategy is computationally expensive for P because of the
large number of feasible nodes that need to be examined.
2. For large-scale models of the form of P, the LP relaxation gap can be reduced by generating lower
bounds, integer cuts, and strong cutting planes. A branch bound strategy with the relaxation gap
reduced performs much better that Benders decomposition for these problems.
3. The lot sizing substructure embedded in the planning problem can be used to obtain nonconven-
tional formulations. Model RP contains more variables and constraints than P. Yet, it possesses a
tighter linear programming relaxation.
4. The large number of constraints in RP can be handled through a constraint generation scheme.
5. Projection of RP onto a lower dimensional space produces PP, a model with fewer variables but
many more constraints.
6. The large number of constraints in PP can be handled through a cutting plane strategy along with
branch and bound, in which only violated constraints are added.
7. Computational results indicate that, for small models, reformulation and projection models do
not provide appreciable gains.
i

Ϫ
␪
it
ء
E
it
ء
ϭ U
i
␶
t
y
it
ء
␶
Յ tϪ
␴
i
␶
t
ء
1if
␪
i
␶
t
ء
0Յ
0if
␪

ء
, t
ء
() ⌰
it
ء
␨
it
ء
Ϫ()
ϩ
© 2001 by CRC Press LLC
8. Straightforward branch and bound becomes extremely computationally expensive for RP and PP
as the problem size increases.
9. For large problems, the constraint generation scheme for RP and cutting plane method for PP
coupled with branch and bound are very efficient.
10. The best method for large problems is to project it in the form of PP and apply the cutting plane
method.
Extensions of the MILP Model
Model P, in the previous section, was developed under several simplifying assumptions. In this section,
we shall show how some of these assumptions can be relaxed to obtain more realistic formulations.
A Concave Programming Model
In model P, economies of scale in the investment cost functions were modeled by the introduction of a
set of binary decision variables (y
it
) to impose a fixed charge on the decision to expand, and a linear cost
function for variable costs. A drawback of this formulation is that, in reality, variable costs are not directly
proportional to expansion quantity. Rather, the investment cost is a concave function because of the
presence of quantity discounts. Thus, a more realistic model for the investment cost would be
(1.34)

bounds and the smallest of the upper bounds, the algorithm may delete any subproblem which has an
associated bound that is larger than or equal to the least upper bound. Details of the method for both
FCP and CCP formulations are presented in reference [10]. Computational results presented in this
paper suggest that the solution of FCP by the proposed algorithm is superior to the performance of
INVT
it
E
it
()
0 when E
it
0ϭ
␤
it
a
it
E
it
b
it
when E
it
0Ͼϩ



ϭ
INVT
it
E

straightforward branch and bound for formulation P. The authors also present means for approximating
the FCP formulation with the CCP formulation, and vice versa, by minimizing the least squared error
of the approximation. Comparison between the solution times of the two alternative formulations indicate
that FCP models are much easier to solve.
Processing Networks with Dedicated and Flexible Plants
Model P was developed for a network in which each of the processing plants produced a set of products
in fixed proportions at all times. Such plants are known as dedicated facilities. Moreover, the plants were
assumed to be operating in continuous production mode. This is usually the case in the manufacturing
of high volume chemicals. However, for low volume manufacture of chemicals, flexible processing plants
are frequently employed. These plants are capable of manufacturing different sets of products at different
times. In general, a processing establishment may consist of either or both dedicated and flexible plants,
which can operate either continuously or in batch mode. Most industrial production facilities involve
continuous dedicated units. For example, paper mills that operate continuously and can produce several
different types of paper are examples of flexible continuous plants. Dedicated batch processes can be
found in the food industry and flexible batch processes are common in the production of polymers and
pharmaceuticals.
The choice among competing dedicated and flexible technologies, and the sizing of each type of facility
to be used, is an important concern at the level of planning the capacity expansion policy of a processing
network. In [15], the authors present an extension of the general model P to take into account the
presence of both dedicated and flexible processing units, which can operate either continuously or in
batch mode. This model is described next.
The chemical processing network is now assumed to consist of two types of processing nodes. Let B
denote processes operating in batch mode and C denote those operating continuously. Let us now
consider the following cases.
Continuous Flexible Processes
Each flexible process i ʦ C can operate at a number of alternative production schemes, each of which
is characterized by a main product. It is assumed that the production rate (r
ijt
) of the main product j of
each such scheme is proportional to the capacity of the plant.

and the main product jЈ,
(1.37)
r
ijt
␳
ij
Q
it
i C jM
i
tʦʦ 1 NT,ϭϭ
W
ijt
␳
ij
Q
it
()T
ijt
i C jM
i
tʦʦ 1 NY,ϭϭ
T
ijt
H
it
i C tʦՅ
jM
i
ʦ

ϭ 1 and Eqs. (1.37) to (1.39) are simplified to
Batch Processes
Here, each process (i ʦ B) is considered separately with no consideration for in-process inventory and
setup costs. Furthermore, no distinction is made between continuous and dedicated units as their
modeling is identical. If W
ijt
is the total amount of chemical j produced by process i in period t and ␴
ij
is the conversion factor for the units of Q
it
and W
ijt
, then the number of batches that can be processed
for product j in plant i in period t is given by (W
ijt

␴
ij
)/Q
it
. Let
␶
ij
denote the batch time for the production
of product j in process i. The constraint that the total available time in unit i cannot be exceeded during
period t is then expressed as follows.
(1.40)
The production amounts of the secondary products in the batch plants are calculated similarly to the
continuous case,
Note that, by letting

T
ijt
i C jM
i
tʦʦ 1 NT,ϭϭ
W
ijt
␳
ij
␪
ijt
i ʦ C jM
i
tʦ 1 NT,ϭϭ
␪
ijt
Q
it
H
it
i ʦ C tՅ
jʦM
i
Α
1 NT,ϭ
W
im
i
t
Q

1 NT,ϭ
W
ijt
␮
ijj
Ј
W
ij
Ј
t
i B jL
i
tʦʦ
jЈʦM
i
Α
1 NT,ϭϭ
W
ijt
␳
ij
␪
ijt
i 1 NP j M
i
t 1 NT,ϭʦ,ϭϭ
␪
ijt
Q
it

and C is a specified constant.
The extended model described is also a mixed integer linear program, and the solution strategies
discussed previously can be applied here.
1.4 Hedging against Uncertainty
As mentioned earlier, many of the parameters of the long-range planning model may be uncertain.
Previously, we assumed that the uncertainties had been sufficiently characterized in the prediction of
these parameters. This section presents ways to deal with the parameter uncertainties more explicitly.
There are two distinct philosophies regarding problems of this type: fuzzy programming and stochastic
programming. Uncertain parameters in fuzzy programming are modeled as fuzzy numbers, while in
stochastic programming they are modeled as random variables with an underlying probability distribu-
tion. Both these philosophies will be discussed and contrasted.
In what follows, first a discussion on the sources and consequences of uncertainties in the model
parameters is presented. Next, the fuzzy programming and stochastic programming approaches are
discussed and a comparison of these two techniques is presented. Finally, we discuss an extension of the
stochastic programming approach known as robust optimization.
Sources and Consequences of Uncertainty
Refer to the original model P. In a general framework, all of the parameters of this model could be uncertain.
However, costs associated with capacity expansions has smaller variability as compared to those associated
with the production, purchase, and sales of chemicals. Also, bounds on the demand and availabilities of
chemicals are very much uncertain, while bounds on the capacity expansions are more or less fixed. Thus,
some of the parameters must be treated as uncertain while others can be considered deterministic.
Once uncertainties are included in P, several problems arise. The planner’s goal is to determine the
capacity expansion plan before the realizations of the uncertain parameters are revealed. If a plan is
obtained by solving the model for one set of realizations (a scenario) of the uncertain parameters, it may
be infeasible or suboptimal for some other scenario. Another approach is to replace all of the uncertain
parameters with their extremal values, thus creating the “worst-case” scenario, and solve the model for
this scenario. Although the resulting plan will be feasible for all other scenarios, it is most often suboptimal
because the probability of such extreme scenarios is usually very small. Such a plan, known as a “fat solution,”
reflects the planner’s complete risk aversion. Also, it is not always straightforward to determine the worst-
case scenario. One could alternatively solve the problem for all possible scenarios and determine which