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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Seasonal Inventories and the Use of
Product-Flexible Manufacturing Technology
by'
Jonathan P. Caulkins
Charles H. Fine
WP#3011-89-MS
May
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
1989
Seasonal Inventories and the Use of
Product-Flexible Manufacturing Technology
by
Jonathan P. Caulkins
Charles H. Fine
WP #3011-89-MS
May
1989
JUL 2 4 19^
J
Inventories and the
of Product-Flexible
Seasonal
Use
Manufacturing
Jonathan
Technology
Caulkins
P.
Charles
H.
August
Operations
Massachusetts
Cambridge,
Fine
1988
Research
Center
Technology
Massachusetts
02139
Institute
of
ABSTRACT
As
advancing
makes
technology
Systems (FMS's) a viable option for
and products, determining their
increasingly
important.
the effect of the
the
ability
Despite
Manufacturing
an increasing number of firms
economic value has become
Flexible
considerable
or inability
to
research
hold interperiod
value of FMS's has received relatively
little
in
this
area,
inventories on
attention.
This paper proposes a stochastic dynamic programming model
production capacity investment decision that explicitly allows
the firm
carry interperiod inventories for safety stock and
to
seasonal stock motives, and it analyzes a simplified convex
A
programming model that considers seasonal stock motives only.
variety of qualitative results are obtained for the case of periodic
market conditions that are not known when the firm makes its
investment decision.
The complete solution is reported on for the
varying market conditions.
periodically
case
of
deterministic
for
the
Examples are given for both cases.
The analysis leads to some surprising results, including the fact
can
flexible
capacity
be
and
that
inventories
interperiod
an
Hence,
the
analysis
can
be
substitutes.
complements as well as
important supplement to unguided intuition.
Inventories and the
of Product-Flexible
Seasonal
Use
Manufacturing
Jonathan
Technology
Caulkins
P.
Charles
H. Fine
Introduction and Overview
1
A
corporations'
low
relatively
of attention
requirements
investment
stifle
procedures
in
investment
this
situation
current
U.S.
for
flexible
in
has received a
that
their
These
investments.
because
to
financial
strict
requirements
can
accounting
cost/benefit
adequately assess the value
to
fail
FMS's
offered
One explanation
attributes
justifying
for
been
of
levels
manufacturing systems (FMS's).
great deal
have
explanations
of
variety
(Michael
of flexibility
and Millen, 1984; Suresh and Meredith, 1984; Kaplan, 1986; Kelly, 1988;
Port
et.
198
al.,
As
8).
a
result,
companies have relied instead on
managerial judgement (Gerwin, 1981; Miles, 1988; Schiller, 1988) which,
without the support of well-designed decision
inadequate (Kaplan,
limited
Progress
on
one
product
Holland,
their
lines
of
type
conventional
that
to
a
at
cycle
sets
Kulatilaka,
competitors
of
often
procedures
in
met with
develop comprehensive models
diverse
a
of
set
time.
Models have been developed
effects
on
value
the
comparison
of
with
the
benefits.
that
to
focus
analyze
FMS's (Hutchinson and
of FMS's that arise from
of
indivisibility
transfer
1986), benefits from being able to produce with either
factor
inputs
1986; 1987),
and
is
quantitatively
accounting
1982; Fine and Li, 1987), the benefits
(Burstein,
capture
to
made, however, by developing models
been
has
life
efforts
tried
because FMS's yield such
success
modularity
two
FMS's
Early
overlook.
have
researchers
of
benefits
the
tools,
1986).
response,
In
support
to
credibly
and
(Kulatilaka
the
ability
threaten
entry
to
Marks,
deter
into
1985; 19 86;
market
markets
in
entry
by
which the
does
firm
not
benefits
related
compete
currently
1986),
and
demand
(Fine and Freund, 1986; 1987;
One
is
inventory
hold
to
used
are
production
hold
to
like
and/or
optimally
respond
to
particularly
unpredictable
decomposing
the
may
Firms
for
at
and
least
investing
analyses.
into
In
interaction
been
analyzing
firm
of
flexible
in
full
constituent
uncertainty,
This
conditions.
generality,
but
by
steps
can
be
parts,
cycle
invest
product-flexible
motives,
stock
motives.
in
safety
Techrrology
have
stock
investment
been
by
studied
at
economics of
the
flexibility
in
Graves (1988) has examined safety stock
have product mix
and
capacity
flexible
However,
flexibility.
seasonal
stocks
has
model
for
ignored.
extends
benefits
the
seasonal
a
product
of
and
and
Fine
the
multiperiod
interperiod
setting
safety
Freund
flexibility
in
which
inventories.
(19
8
to
7)
hedge
the
firm
Section
2
and seasonal stocks.
The
rest
programming model which excludes
on
analyze
seasonal
the
when
stock
hold
presents
for
a
both
of the paper analyzes a convex
safety
Section
stock
3
considerations
presents
six
and
theorems
when market conditions are seasonal but
firm makes its technology investment decision.
model
the
issues.
against
can
dynamic programming formulation which allows
stochastic
would
capacity
(1987) and Vander Veen and Jordan (1987),
facilities
in
uncertain
market
considerations
stock
addition.
uncertainty
that
in
and/or
reasons:
stock
between
paper
This
focuses
the
with the option
firm
a
non-stationary
its
inventories
three
Kekre
when
requirements
safety
value
the
flexible
in
addressed
although they do not look explicitly
largely
that
is
it.
hold
cycle
Karmarkar and
the
future
Since production and
affect
know how
to
be
to
seasonal
and
decisions
their
to
variability
variability
toward analyzing
motives,
about
al.,
1988).
models
inventories.
dynamic,
complex
too
is
capacity
et.
meet demand, intuitively one would
to
inventory
taken
market
capacity.
holding
problem
uncertainty
these
all
1988),
1986; Roth
He and Pindyck,
inventory
Ultimately one would
of
against
interperiod
together
ability
the
hedge
to
assumption of
restrictive
not allowed
expect
ability
Pappu,
with flexibility (Gaimon,
associated
the
and
(Fine
Section 4 reports on a complete solution for the case of deterministic,
periodically
known with
unlikely
to
studying
this
Examples
results
Market conditions are rarely
varying market conditions.
are
so
certainty,
be
applied
solution
given
in
and conclusions.
the
directly
provides
both
solution
in
practice,
insights
Sections
described
3
about
and
4.
in
Section
but
we
the
general
Section
believe
5
4
is
that
problem.
discusses
The General Stochastic Demand Problem
2
Problem
2.1
Description
The problem considered here
firm
must make
face
of
can
about
uncertainty
capacity
investment
market
conditions.
future
switched
be
period's
production
the
produce
any
capacity
requires
of
resolved
before
is
to
the
N
have
the
in
The firm can
products
larger
a
find
the
optimal
capacity and
its
trade-off
but
sells,
it
investment
initial
into
all
completely
This
makes
firm
after
committed
is
it
that
facilities
does
than
unit
problem
the
can
production
flexible
per
that
is
to
between the benefits of flexible production
higher investment cost.
subsequent
single
a
Each
products.
the
The crux of
Fine and Freund (1986, 1987) address
when
N
the
production
flexible
capacity dedicated to a single product.
production
period.
suppresses
and
sales
this
activity
This simplification yields
the
issues
posed
problem
by
can
useful
for the
be
collapsed
results,
interperiod
case
but
it
inventories.
paper extends their model to an arbitrary number of production
sales
period
2.2
decision
a
Because of the uncertainty,
production capacity.
like
of
among any
and inventory decisions but
in
would
firm
cost
uncertainty
investment
its
zero
at
demand
and
zero,
a
in
period's
to
irreversible
At time
following.
mix of non-flexible production capacities each dedicated
one of its N products and/or more expensive flexible capacity that
invest
to
an
the
is
and
periods
into
the
allows
the
firm
subsequent period.
Formulation
The formulation uses
the
following
indices:
>^(1
N}
indexes the product,
se{l,...
,S)
indexes the
tG(l,... ,T)
parameters:
state,
indexes the f>eriod;
and
to
carry
inventory
from any
a =
r
=
the per period discount rate and
purchase
the
product
one
of
cost
and rp
i,
vector of capacity purchase costs,
the
is
[ri,r2,...,rN,rF]
capacity
of
unit
T[
dedicated
is
to
the purchase cost of one unit of flexible
is
capacity;
variables:
K
= [Ki,K2,...,Kn,Kf] is the vector of production capacities, K;
the amount of capacity dedicated to product i, and Kp
amount of
the
period
period
product
of
produced on dedicated capacity
in
produced
in
i
on
flexible
capacity
t,
= amount of product
lit
i
t,
amount
Zit=
is
capacity,
flexible
= amount of product
Yit
is
held in inventory from period
i
to
t
period t+1, and
<Jt
and
= the
state in period
functions:
= revenues generated by selling x units of product
Rit(x)
state
= cost of producing x
hit(x)
= cost of inventorying x units of product
It
which
assumed
is
on the time and
the
in
state
that
s
the
is
future
of the
state
in
s'lot
=
s)
period t+1 given that
determined
example, Y,
=
noting
by
0,
and for
Y^j e
Yii, Y,2
inventory
t
levels
i
from
and
period
a
market depends only
=
as
1,
...,
a
is
in state s in period
SR
are
For
.
K,
the
vector of capacities chosen
T, pt(s,I[_i,K)
function
t.
The dimension of vectors can
which subscripts have been omitted.
The decision variables
time
in state s,
the probability the market
is
it
Boldface will be used for vectors.
be
i
current state and not on the firm's past decisions.
its
s'
units of product
realized to the next period.
formulation, Pi(at+i =
will be in state
in
i
s,
Cit(x)
in
So
t;
of
the
the
period
current
inventory vector, and the vector of capacities.
t
production
state,
the
at
and
current
The
following
General
K.p,(s.I,.K)
t
Demand
Problem
dynamic programming formulation:
V
Max
Stochastic
= l...,T
'OO'^O
N
= -Xr.K, - ffKf
,=1
(GSDP)
has
the
the
In
H
sequel,
*
and the superscript
variables
capacity decision
denote the vector of
will
production
all
and
denote optimal
will
values.
Since
any
at
given
time,
demand
future
market
motives
as
can
be
However,
well.
of
function
a
this
numerically because the solution
difficult
to
obtain
analytic
To our knowledge,
the
that
interaction
of Graves
safety
would
model.
goal
(198
relative
desire
only
8).
to
The work
That
required
to
one
analyze
in
the
with
both
is
large,
stochastic
work
between product mix
inventories
flexibility
for
is
that
a
safety
remainder of
this
general
solve
it
is
made progress on
and
safety
the
stocks
difference
is
in
mix
Ultimately
one
effects
in
one
paper moves toward
this
with
flexibility.
and
in
to
stock
complete
system
no mix
this
dynamic programs.
has
flexibility
difficult
and
work characterizes
for
seasonal
captures
it
formulation
space
results
the
time,
uncertain,
Since the state of
formulation clearly includes safety stock motives.
the
is
seasonal
by analyzing the interaction between seasonally varying market
conditions
and flexible manufacturing capacity.
3
The Unknown Seasonal Demand Problem
3.1
Formulation
The General Stochastic Demand Problem (GSDP) considered in
previous chapter is too complex to yield to tractable analysis.
the
However,
functions,
it
revenue
the
We
functions
thereby
periodic,
programming
model
that
Demand Problem (USDP).
This
we
formulation
cost
result
Unknown
requires
this
stock
safety
The
the
label
and
accomplish
excluding
focusing on the seasonal stock problem.
issues and
convex
on
structure
can be simplified into one that does.
making these
by
more
imposing
by
is
a
Seasonal
following
the
assumptions.
The revenue, production
Al:
=
Ot-i
si
holding
This can be modelled
periodic with period two.
Pi(at =
and
cost,
r) =(
1
1
if s
is
if s is
odd and
even and
=
s
s
=
functions
are
conditions
for
cost
as:
-
r
r
1
+
1
otherwise.
However,
more
is
it
natural
identify
to
the
market
both odd and even periods with a single state and the time subscript
t.
Then
A2:
the
= C-i(x),
Rit(x)
and
h-j(x)
=
h-i(x)
Cit(x)
= Ci2(x),
R[t(x)=Rj2(x), and
h[t(x)
=
h-jlx)
Rji(x).
state
s
is
for all
when the technology investment
However, the firm learns
unknown.
sets
of functions,
Assumptions Al
before production
periods.
(1)
demand,
This
is
inventory
(2)
and A2
at
even
and
t,
t.
undertaken,
is
the
state,
and
beginning of period one.
the
imply
odd
for all
At time zero,
therefore both
that
=
C-t(x)
that
all
uncertainty
is
resolved
and inventory decisions are made for any of the T
a
strong
policies
technology
assumption,
can be used
investments
but
to
it
captures
respond
must
be
to
the
key ideas
uncertainty
made
before
in
the
of
resolution
and
production
inventory
and
investments,
policies
holding
(4)
uncertainty
after
(3)
are
resolved,
is
constrained
by
inventories
can
interperiod
the
sunk
the
affect
the
of flexible capacity.
utility
Periodicity
common.
in
uncertainty,
this
and
inventory,
production,
in
conditions
sales
is
For example, producing and transporting bulk commodities
Lakes Region
the Great
more expensive
is
the winter than in the
in
Likewise, holding
summer because the Inland Waterway is closed.
For U.S. automobile manufacturers, holding
costs may be seasonal.
cars from summer to winter is more expensive than holding them
from winter to summer because many new models are released in
the fall, reducing the value of cars held in inventory from summer to
the
company
for
a
less
can
Cost differences
winter.
toy
of
from
arise
not just from weather.
itself,
manufacturer may be lower
its
Christmas
seasonal
warehouse
space
Seasonal
Season.
For example, holding costs
spring
in
devoted
is
within
variations
in
toys
demand
in
because
fall
stockpiling
to
variations
than
the
for
extremely
are
common.
The subsequent analysis depends on
the
following
assumptions
about the capacity costs, revenue functions, and cost functions.
assumption about production costs
the
in
is
ever likely
Only
be restrictive
to
practice.
A3:
The
revenue
functions
differentiable
argument
A4:
The
argument
cost
differentiable
is
zero.
quantity
they
bounded,
that
are
nondecreasing
strictly
concave,
when
their
of each
convex
functions
Cit(x)
functions
are
are
that
strictly
finite
increasing,
when
their
Production costs are a function only of the
which kind of capacity
Strictly
are
zero.
production
convex,
total
is
functions
Rjtlx)
product produced
is
in
the
not
period,
of
used.
production
imply diseconomies of
costs
scale,
10
are
but
relatively
linear
rare
because
production
costs.
which
common,
reasonably
are
The
formulation.
assumption
are
that
reasonable
The holding
convex,
functions
cost
our
costs
are
when
Note,
if
argument
their
that
inventory
levels
are
strictly
nonnegative
and
are
zero.
is
uniquely
are
sales
increasing,
strictly
holding cost functions
the
is
costs.
are
h,i(x)
functions
differentiable
finite
convex,
A6:
production
in
example, raw material costs are the dominant
for
if,
component of variable production
and
unit
encompassed
and roughly equal for flexible and dedicated capacity
linear
A5:
also
are
determined,
convex but not
but
not
strictly
production
the
separately.
Production
in
advance of demand and storage costs more than
production
in
the
subsequent
period.
N
A7:
rp >
r,
>
for all
i,
rp< ^r,
but
i=l
were
If rp
never
A8:
than
less
equal
or
economical
be
i.
sum
firm
of the
the
ri's,
flexible
capacity.
(R,[W -
Cij(x)) is
This
is
is
a
ri
in
some
to
i,
then
purchase
it
would have no incentive
nondecreasing
at x
the proof of the
=
to
would
capacity
were greater than or equal
If rp
technical condition which
used only
for
firm
the
for
dedicated to product
to
to
the
purchase
0.
would generally be
true
and
uniqueness of the Karush-Kuhn-
Tucker (KKT) multipliers.
With
models the
greatly
these
eight
seasonal
simplify
the
stock
assumptions
but
issues,
analysis.
1
1
the
three
formulation
reasonably
more assumptions
will
)
The
A9:
N
=
analysis
restricted
is
two-product-family model; that
to a
is,
2.
This assumption greatly facilitates explication of the model and
has applications in a number of settings.
To
198 7.)
distinguish
products
are
capacity
will
subscript
'F'.
labelled
be
'A'
the
product
and
'B'
is
All: The
inventory
can
firm
carry
period
indices,
Also,
sequel.
subscript
the
AlO: The planning horizon
and
the
in
by
identified
(See, e.g.. Fine and Freund,
the
the
flexible
instead
'AB'
two
of
the
infinite.
into
the
first
period
a
at
cost
equal to the production cost in an even period plus the holding cost
from an even period
AlO
Assumptions
with
associated
They
with
exploited
extensively.
boundary
effects,
up
building
With
(ordered)
the
another way,
the
production
Letting
j
a
the
the
boundary
complicating
desired
of
level
time
infinite
effects
transient
inventory.
set
set
even
of
of odd
all
the
and
even periods or
periods
inventory
levels
are
the
is
and
no
cannot
be
Or,
periods.
"after"
that
horizon
odd periods do not necessarily have
coming "before"
periods and for
to
an
from the (ordered)
optimal
and
period
distinguished
of as
eliminate
symmetry between even and odd periods
create
also
All
and
terminal
a
associated
effects
an odd period.
to
to
put
it
to
be thought
them.
Thus, the
same
for
all
odd
even periods.
be the index identifying odd and even periods
e
{1,2}
ps
= P(ai =
_
and ^=77~ii
1
respectively,
Demand Problem (USD?)
Max V = -
r^
Ka -
rg
Kb -
^i Ps i I
s=l
(
s),
,
the
Unknown
Seasonal
is:
r AB
«:.(
Kab +
y:,. z' * .;- i:J -
i=Aj=l
12
c:,(
y:,. z;j -
h:,(
i:j
subji
:
Lemma
In
(i)
1^2=0
3.
:
1
any optimal solution Z, for
That
both.
or
is,
(i,s),
=
l'i
or
never optimal for the firm to carry
is
it
product/state pairs
all
inventory of any product in both odd and even periods.
s
(ii)
2
S
2
/
s
s
S=l
is
some
some
s
pair
/
That
.
for all products there
is,
which
in
uses
firm
the
available
all
And,
if
Y
there
and
is
be fully utilized.
will
See Caulkins (1988).
USDP
The
3.1
Actually
Z
has a unique optimal solution.
not
are
surplus
capacity
the
firm
determined
uniquely
necessarily
is
indifferent
producing with dedicated or flexible capacity, but their sum
is
uniquely
of
K, X, and
determined,
I
is
because of
and,
indifference,
=
Y
+
Z
knowledge
See Caulkins (1988).
The next theorem gives formulas
Tucker (KKT) multipliers.
It
which
all
are
USDP
is
the
this
X
between
essentially a complete solution.
Proof: Straightforward.
KKT
Ka +
\
s
Zaj + Zbj
state/period pair, production capacity
Theorem
the
and
/
j=l
both dedicated and flexible, to produce that product.
Proof: Straightforward.
because
\
J=l
state/period
capacity,
in
s
s
\
s=l
Kb + Kab = Max Max \Yaj+ Ybj+
/
Kab = Max Max Yy + Zy
In any optimal solution H, Kj +
constraints
strongly consistent;
conditions
given by
sufficient for optimality.
for
the
Karush-Kuhn-
optimal
possible to find a feasible solution for
is
satisfied
i.e.,
I
it
-
as
satisfies
strict
so
inequalities,
Slater's
condition.
the
Thus
VII below are both necessary and
Primes denote derivatives.
2 and vice versa.
14
If
j
=
1
then
j
=
Necessary and Sufficient Conditions for Qptimality:
i
(II)
Y
p,[r^'(y;,+z;,+ i;^-i;)- c^'fY^+z;,)) =
i
(III) Y
p,fRj'(Y^z;+i^-i;^) -
Ki'^l^K^K-^^
(IV)
= A.B;
h^'( Y-
= 5-j-q'j
= A,B;
j
=
1.2; s
=
1..S
y;-5;-uJ
j
=
1,2; s
=
1..S
j
=
1,2; s
=
1..S
+z;)i
= A,B;
s
sjj
—
=-
YPs|M^^\R..;:-h
/
- Cy
j
=
U;
1.2:
s
=
1..S
i
= A.B;
j
=
U;
i
= A.B
= A,B;
i
=
j
—
_
5
=
^.
=
m.AB
-
r:
s
=
1..S
s
=
1..S
/_s\
B
5
Rn
_s
:
ZSc^ij
X ZY
-
r..vB
j
s=i j=l
See Appendix.
Proof:
These
have
multipliers
The multiplier
interpretations.
a.,
shadow value oi having another
i
available
value
in
of
Similarly
state period
corresponding
the
the positive
is
5..
whereas
multiplier
pan
firm
should
dedicated
as
to
willing
product
of
the
period
to
negative
j
to
pair
pay
so that in
i
Finally,
(s,j).
to
'upgrade'
state/period
u-j
one
pair
able
in
The
it
unit
the
is
to
state
part.
of
amount
of capacity
unit
(s,j)
value.
can be used
one unit of flexible capacity.
Suppose
dedicated
values
A
be
shadow
shadow value of having an additional
the
is
absolute
the
is
the
j
product
to
shadow value of being
from period
value
absolute
s^
of
part
of the
i
dedicated
Multiplier
negative
capacity in state/period
flexible
the
y.
the
is
q..
of capacity
(s,j).
inventory one more unit of product
s,
simply the positive part of the
is
unit
pair
economic
straightforward
in
Ka'
that
capacity
of having
optimality
the
product
exceeds
at
for
an
additional
each period and each
=
0.
dedicated
In
capacity
for
unit
Then by
is
the
product
.A
must
1
6
the
mA
sum
of
dedicated
>
fall
before
hold for
price
of
the
shadow
to
product
and thus by VI,
0,
amount by which
Similar results
Kb and Kab-
I'V,
purchase
unit
of capacity
mA
other words.
purchase that technology.
respect to
state.
.A.
per
it
mg
is
price
of
optimal
to
the
and mAB. with
These interpretations of
left
hand sides of conditions
of condition
restriction
whatever
that
conditions
and
II
product
to
can
statements
Similar
be
to
be
concept
optimal
of
period
in
i
produces
it
made
to
product
must
about
each
purchase
flexible
The
solved independently.
Max V =
subject
-r. K, -
decomposes
subproblems
Subproblem
^tpsl
variables
all
sides
of
consider
A
and
two
convex
are
for
i
proof uses
Its
these
B
is
the
subproblems
their
sharing
i
=
i;,)
A
or
one
subproblems.
programs
B
that
for
can
is:
- C^jIy^j)- h^(l*)!
S
=
j
=
j
= l,2;s=l..S.
KKT
=
1..S.
conditions are:
C^'(
Y^))
2
SS«ij
1.2; s
nonnesative.
Yp,(R^'(Y^+I;--I*) -
ri
j.
to:
The corresponding
(III)
period
hand
left
the
and sufficient condition for
into
R^( Y^+l]^-
I^- Y--i;-<0
(I)
in
to
firm does not purchase flexible
the
If
Yy -K, <
with
the
hand side
subject
is
sold
capacity.
We
subproblems.
problem
product.
be
the
necessary
a
of flexible production capacity.
the
that
j
because the only linkage between products
capacity,
left
with
III.
The next theorem gives
it
For example, the
III.
-
I
consistent
are
gives the marginal value of having an additional unit of
I
dedicated
capacity
multipliers
the
+ mi
5=:j=i
17
=ay-5j-sj
j
=
U:
s
=
1..S
j
=
1,2; s
=
1..S
be
(IV)
a;j(K.-Y;j)
=
j
(V)
,s
s
mi Ki =
s
(VI)
s
s
s
aij,5ij, qij.Sy,
and mi are
all
nonnegalive.
=
The
a
hand side of
right
of flexible
unit
firm should invest
than
less
is
shows how
without
capacity
solutions
much
Denote the
Theorem
3.3
and multipliers
m^B
problem
and
Theorem
in
3.1
r^s^
implies
the
satisfy
m^
multipliers
the solution
<
is
The formulas
S.
subproblems
0,
K^b =
a
solution
satisfy
that
is
y
and
the
full
u,
0,
to
the
all
so r^B ^
unique,
Let r =
the
formulas of Theorem
3.2.
K^i^
KKT
S implies
r^s^
and
{t^, rg, r^a)
V*(r)
(ii)
V*(r)
That
acquiring
is
V*(r)
capacity
The
convex
is
= -
the
K
r'
+
for
the
But since r^B
optimal
the
*^
S,
KKT
Since an
not optimal.
< S implies K^^b* >
denote
0-
value
of
problem
f(E).
in r,
nonincreasing in
is
^yJA = -
(ill)
earn.
multipliers
3.4:
(i)
of
0,
3.2 yield a set
which contradicts the nonnegativity of
<
as a function of r by V*(r)
Theorem
=
Theorem
in
then
If
optimal solution always exists,
can
Z =
yields
Hence, r^^ < S implies K^^ =
multipliers.
costs
This
3.2.
KKT
of
set
multipliers.
subproblems
USDP
both
subproblem
problem.
full
subproblems with
the
suppose
optimal
this
the
0.
Now
of
the
it
flexible
in
Only
solving
generally
because
of investing
problem.
full
the
marginal cost
its
useful
is
It
benefit
if
that
hand side of the inequality in the statement
Suppose r,j^ > S.
Then augment the solutions
S.
for
a
benefit.
asserts
right
By Theorem
conditions.
Kab* =
by
defined
as
and
solving
than
easier
the
simply
and only
if
marginal
the
solving
needed,
are
Proof:
of
measure
to
marginal
marginal benefit of
the
is
theorem
this
capacity
flexible
in
expected
the
so
capacity,
condition
this
K,
is
for
i
and
= A, B, and AB.
nonincreasing in r
acquisition
convexity
advanced
r,
increase
of
V*(r)
manufacturing
19
is
the
quite
maximum
suggests
technology
decreasing
intuitive;
that
profit
as
continues
the
to
the
firm
cost
of
decline.
firms
tells
enjoy
will
exactly
large
increasingly
how
improvements.
profit
Part
(iii)
rapidly profits improve as capacity acquisition costs
decline.
Knowledge
reductions
V*(r)
of
acquisition
in
useful for evaluating potential returns to
is
and because there
costs
often
is
uncertainty
about these costs, so sensitivity analysis with respect to the capacity
costs can
be informative.
Theorem
3.4
describes
behaved function, so
form
its
and
guarantees
that
but
difficult,
is
well-
a
is
it
reasonable to draw inferences about V*(r)
is
it
obtaining V*(r)
general,
In
from points obtained numerically.
Proof:
Proof of
Let (K,Z)
Since (K,E)
V*(r2) > -
be
K
+
Proof of
= V*(?cri + (1-X)
Let
(ii).
r
let
[0,1]
vectors
cost
= Xti + (1-X)
corresponding
solution
-
ri^
K
to
r.
+ f(E) and
ri
and vi
Assumption A7 such
that
V*(r)
r2), so
be
ri
two
is
convex
capacity
> ri and ri ^ r2.
in r.
cost
vectors
Let (K,2) be
Then V*(ri) = - ri'
V*(r2) > - r2' K + f(S).
But
unique optimal solution corresponding to ri.
the
K
optimal
capacity
Thus XV*(ri) + (l-X)V*(r2) > - (Xrit + (1-X)
f(E).
r2')K + f(E) = V*(r)
satisfying
two
For arbitrary X e
unique
the
be
feasible for all cost vectors, V*(ri) >
is
r2'
and ri
Let ri
Assumption A7.
satisfying
r2.
(i).
+
{(E).
Since (K,Z)
is
feasible for rj,
then V*(r2) - V*(ri) > (ri'-
Proof of
(iii).
theorem (Varian,
This
r20K
is
>
0,
so V*(r)
direct
a
nonincreasing in
is
application
of
the
r.
envelope
1978) as extended to nondifferentiable functions by
Fine and Freund (1987).
3.3
Case of Quadratic Revenue and
Holding Cost Functions and Linear Production Costs
Analysis
of
Special
As discussed above, knowledge of V*(r)
reasons.
For a general
USDP, V*(r)
but even more can be said
when
the
is
is
a well-behaved function of r,
USDP
has quadratic revenue and
Theorem
holding cost functions and linear production costs.
an exact, second order Taylor expansion for V*(r
shows
that
the
optimal
values
20
of
valuable for several
the
-t-
Ar).
decision
3.6 gives
Theorem
variables
3.5
are
piecewise
continuous,
of
case
USDP,
the
function,
solution,
entire
the
USDP
Theorem
3.5:
functions
and linear production costs,
are
vector
r,
the
optimal
the
value
all
linear
that
statement
decision
3.6:
K
function
the
in
is
from the
directly
quadratic
a
is
M
is
optimal
the
dr^
dr^
capacity
and the
first
The
first
programming
unique.
is
for
vector,
which V*(r)
then
is
Ar
for
drg
drg
drg
where
= -
positive
^^AB ^^AB
semi-definite
Proof:
Formula
positive
semi-definite
av*(r)
piecewise
function.
of a region
interior
dr^
^AB
and
capacity cost
of the
V*(r+ Ar) = V*(r)-rAKA-rBKE-rABKAB + -Ar'MAr,
1
M
is
small,
sufficiently
(1)
If r is
and
quadratic
r
optimal
variables'
continuous,
a
is
and Theorem 3.1 which guarantees the solution
Theorem
dependence on
its
consequence of the theory of quadratic
a
is
value
r.
objective
the
and
special
this
optimal
the
just
functions
function
The second statement follows
Proof:
not
for
with quadratic revenue and holding cost
piecewise
continuous,
function of
quadratic
fact
For
and
r,
Hence
r.
simple nature.
of a particularly
values
of
well-behaved function of
a
is
functions
linear
= -K,
for
i
(1)
follows
directly
because
V*(r)
from Taylor's Theorem.
is
convex.
= A,B, and AB, where K,
is
the
dli
a
Hence, M^j =
v*(r)
arjar.
a /av*(r)
ar.
^(-K,) =
ar.
ar,
21
-^
ar,
By Theorem
optimal
M
is
3.3,
capacity.
3.4
Example
This
and
functions
gives
section
unitary
maximum
volume.
a
demand.
inelastic
per
If
assumed
costs will be
Revenue
to be
functions
concave
are
Assumption A3.
generated
Similarly,
not
1,2
}
unitary
concave,
holding
linear
cost
they
so
functions
not guaranteed, but that
is
if
Yjjj.
= Rik ~
volume
sales
j
hjj
if state
s
is
not a
demand can be
for product
i
in
period k
if
i
in
period
j
and selling
j
and sold
X Y^,,
z;,
and sold
=
i
if state
s
is
j
to
realized,
produced on dedicated capacity
in period k if state s is realized,
i
in period
I Z^k,
period k
in
it
be the amount of product
j
from period
be the profit from producing one unit of product
in period
period
i
realized,
be the amount of product
k=l
is
state s is realized,
in period
Zjjij
a result,
realized,
is
s
period
i
convex
Let
be the cost of carrying one unit of product
^ijk
violate
are
As
be the constant revenue per unit sold of product
k
are
demand
inelastic
so they violate Assumption A5.
maximum
be the
Rji^
costs
be the index of the sales period,
state
hij
holding
with linear costs and unitary inelastic
restated as follows.
Djjc
and
some
to
example.
this
USDP
{
by
strictly
uniqueness of the optimal solution
k €
sold,
zero.
but
but not strictly convex,
The
up
unit
Hence, without loss of generality, production
holding cost functions.
concern for
per
production
unit
USDP with linear cost
A firm facing unitary
production costs can be absorbed into the revenue and
then
constant,
So Y^ =
of
demand earns constant revenues
inelastic
curves
example
an
and
k=l
22
![
and
produced on flexible capacity
k
if
state s is realized.
= Y^^- +
Z^.-.
in
The objective function can be
V = -r'K
+
=
The maximum
II
I
XPs
s
written
'^.jklvjjk+z;
(y^.+ z;J
i=A j=l k=l
l
imposes the following constraint:
condition
sales
as:
^iik + Zji,^ + Yi2k + Zi2k ^ Dik for
in addition to the usual constraints.
This
such
techniques
program,
linear
a
is
simplex
the
as
so
all
can
it
and
i,k,
solved
be
method.
s
can
It
demand
remaining
yielding
triplet
capacity
from
yielding
positive
a
one
If
lists
formula
general
Z.jk
K-AB-
where
has
U(7Cx, T^y)
Ky or Kx =
extent
greatest
possible,
the
unit
profit
either
until
all
(1)
and
TCy
been
allocated
the k's
descending
in
solving
for
linear
the
cost
all
(2)
a
k^.
order
second
functions
problem with
Zijk),
U(7rijk, 7tijk)(Yijk
+
Z,jk)
Ki -
rty
-
}
Yjjk,
U(7tijk, Kijk)Zijk
in the list
unitary
U(7tijk, :tijk)Yijk
-
U(7tljk, 7tijk)Ziji^}
modified unit step function.
precedes
a
is:
U(7iijk, JrijkXY.jk
-
then
of profitability,
stage
+
U(7:,jk, rtijk)Zijk
is
or
profit.
= Min{ Dik = Min{ D,k -
Yijk
per
greatest
demand and
inelastic
the
One
trivial.
is
product/production-period/sales-period
the
for
the
to
solved
demand has been
product/production-period/sales-period
combinations
production
met
meeting,
continues
standard
be
also
numerically because solving the second stage problem
simply
by
of
U(7ix. T^y)
=
1
if :tx
>
Otherwise, U(;rx.
Jtijk's.
Ky) = 0.
Example
nature
The example
of
relationship
it
considers
the
(parameterized
uncertainty
between
about
inventories
two extreme
and
situations:
by
market
flexible
(1)
^)
considers
conditions
capacity.
when
the
firm
how
affects
the
the
Specifically,
knows with
demand for each product will be, but it does not
know how seasonal the demand will be (^ = 1) and (2) when the firm
knows with certainty what total demand for each period will be, but
certainty
it
=
what
does not
total
know how
it
will
be divided between the two products (^
0).
23
The underlying demand
product
However,
random
each
in
actual
the
with
variable
and
period,
assumed
is
demand
total
demand
be for one unit of each
to
is
always
product/period
in
pair
four
units.
(A,l)
a
is
following distribution:
the
x
Pai
=
1-2Pai x=1
x
Assume
for
moment
the
=2
these
that
random
two
variables
are
independent.
Furthermore,
let
^ ^ <
1
be the fraction of the change
demand for product/period pairs (A,l) and
same product but the other period, and
comes from
the
the
variability
is
between periods.
same period but
3.1
- ^ be
1
the other product.
between products, and
Table
(B,2) that
if
^
=
1,
Table 3.1
Description of
s
the
Thus
all
fraction
if
^
=
Demand
States
the
that
0, all
the variability
summarizes the demand distribution
each of the nine possible states of the world.
State
comes from
in
is
in
between
when
happens
what
consider
First
=
^
so
1
Similarly,
pairs.
product/period
and
states
and
2(Ka +
Kg)).
relative
to
over
both
holding
- 2(Ka +
second
stage
assume
costs
and both products
meet demand if K^s^^ ^
under each state
r^B >
(Specifically,
idle
-
[1
how much
are
large
- 2Pai)(1 -
(1
i.e.,
K^^ =
inventory
be
will
s.
(fab - ta - 2
fAB < ta + 2 PaiPb2
Likewise,
optimal.
Kg =
otherwise.
1
Hence, for
holding
the
(l/2)(4 -
revenues
and
holding
of
part
the
costs
function
objective
to:
Max V =
if
in
easy to solve the
is
it
do not depend on the decision variables, the
reduces
for
four
is
costs
any purchased capacity;
determine
to
constant
the
investment
capital
With these assumptions
Kg)).
problems
Ignoring
that
all
always
firm
the
periods
firm can always
Finally,
(l/2)(4
So,
ensure
for
so they can be omitted from the objective
2PB2)]h) so the firm does not
held
to
equal
be
sold
unit
enough
high
be
solutions,
all
so the
state,
per
product/period
all
With these assumptions revenues are the same
Demand
function.
revenues
let
pairs
meets demand.
every
is
periods.
Let the per unit holding costs h be equal for
all
variability
all
costs
cost
of
h,
PaiPbz h)KA +
- 2 PaiPb2 h)KB.
rfi
is optimal.
Otherwise, Ka =
is
Ka =
is optimal;
^b + 2 PaiPbz h, then Kb =
then
tab <
if
(tAB "
1
is.
problem and these parameter values, increasing
this
favors
investment in flexible capacity,
and decreasing
reduces
amount
capacity
flexible
expected
the
So flexible capacity and inventory are
inventory.
however, that even
if
nine states inventory
only flexible capacity
is
still
This
used.
is
is
of
Note,
substitutes.
purchased, in six of the
simply because no amount
of flexible capacity can shift a unit from one period to the next; only
inventory can.
favors
then
Also,
investment
Kab =
is
purchase
flexible
variability,
capacity.
optimal no matter
how
increasing Pai or Pb2.
i.e.
(However,
large
Pbz
that
if
is,
if
Pai ^
for
were positively correlated the firm would be
less
flexible
one
would
capacity
expect
and,
conversely,
25
if
—2h—
and vice versa.)
demand
Intuitively
products
in
increasing
their
two
the
likely
demand
to
were
5
negatively
correlated,
This intuition
flexible capacity.
Suppose P(s = 6) = P(s =
PaPb +
Then
£•
demand
for
function
is
Now
will
The firm knows
2e
=.
—
=
in
for
A
and
The objective
r=
and
stockout
meet demand under
the
possible
there
all
variability
what demand
will
the
severe,
is
each
in
divided
be
or
high,
sufficiently
is
firm
will
long as Pai and Pb2 are both
^ 2-Min{KA, Kg} because it is
of
as
demand
optimal
not
is
it
it
capacity.
flexible
one product.
always optimal to have Ka =
be four units
and
2,
Then
and
inventories,
so
0,
are
sufficiently
are
implies that
this
revenues
If
states.
=
^
certain
for
must have Kab
will
and Kab =
B.
costs
all
firm
rg,
2(PaiPb2 - e)h)KB.
rfi-
know how demand
does not
it
A
products
nonzero,
extreme,
other
the
but
be,
equivalently,
0,
P
is
demand favors larger
demand favors investment in
consider
between products.
Tab < ta +
demand
coefficient between
correlated
correlated
negatively
between
invest
to
and P(s = 5) = P(s = 9) =
e
= (rAB- ta- 2(PaiPb2 - e)h)KA + (rAB"
positively
season
likely
now
Max V
So,
= PaiPb2 -
same period
the
in
more
be
correct.
is
8)
correlation
the
B
would
firm
the
Since
for
inventory
carry
to
0,
Kg =
even
if
holding costs are zero!
Clearly
case
this
The only way
capacity.
be
in
demanded
is
to
inventory
A can create
2-Min{KA, Kg} is
stockout
costs
converts
one
converted
initial
are
not
product
into
dollar
substitute
for
flexible
large.
which
In
a
sense,
Shortage
another.
losses
of
even a single unit of product B.
not necessarily optimal, however,
arbitrarily
into
will
No amount
capacity.
flexible
inventory of product
Kab ^
a
meet uncertainty about which product
purchase
to
not
is
can
be
of
one
recovered
stocking
out
product
by
if
is
reduced
outlays or increased sales of another product.
example suggests that insight can be gained by
decomposing total demand variability into variability between
This
products
suited
and
for
variability
meeting
between
variability
periods.
between
26
Flexible
products
capacity
and
is
best
inventory
is
best
the
suited
meeting
for
objective function
by either product
which
flexible
periods
in
variability
measured
is
either
capacity
and
inventory
this
example,
between periods.
in
period,
stocking out provides
compensate
can
which can be generated
dollars,
compensate
can
However, since
means by
variability
between
variability
between
for
for
a
products.
In
negatively
investment
how
this
safety
correlated
in
flexible
stock
both
increasing
demand
capacity.
for
This
considerations would
paper for seasonal
stock
variability
the
general
products
and
favored
matches intuition and suggests
modify
considerations.
27
two
in
the
results
obtained
in
Known
4 The
This
section
one
results
KSDP
identical
is
for
the
Known
of
the
USD?
case
special
a
vary periodically,
conditions
of the
reports
(KSDP),
Problem
Demand Problem
Seasonal
but
are
USDP
which
in
market
The formulation
deterministic.
of the
to that
Demand
Seasonal
except that there
state.
Market conditions are rarely deterministic, so
model
applied
be
will
practice,
in
nonetheless.
For example, we observe
investing
flexible
in
can
capacity
useful
gives
but
it
that
(1)
insights
holding inventory and
complements,
be
unlikely this
is
it
(2)
can
it
optimal to purchase less capacity dedicated to a product than
in
any period, and (3)
Section
if
gives
4.1
not,
expressions
revenue
case
is
KSDP.
in
called
some
optimal
the
functions
that
always hold.
fact,
for
are
analysis
sensitivity
how
are
Seasonal
(1988).
is
when
variables
This
linear.
special
Demand Problem or Qsolutions to the Q-KSDP.
derived.
Section 4.4 gives
results.
results
were compressed sufficiently
informative.
one might expect to be
decision
the
costs
solution
The derivations of these
4.1
of
and
KSDP
the
to
possible to obtain closed form
is
It
Known
the
sold
constant.
solutions
'properties'
Section 4.2 gives properties of optimal
Section 4.3 describes
be
known and
is
of optimal
values
quadratic
Quadratic
the
demand
its
is
be
be optimal to use flexible capacity to
properties
and shows by example
do
may
it
produce a product even
true
just
is
They
are
to
are
present
reported
in
quite
here,
their
involved,
they
and
if
they
would no longer
entirety
in
Caulkins
Interested readers are invited to request a copy.
Properties
of
Optimal
Solutions
The following statements about production and inventory hold
for all optimal solutions to the
1)
If
it
is
optimal to purchase flexible capacity,
used to produce one product
the
other
KSDP.
in
it
will
all
one period and the other product
period.
28
be
in
For
2)
acquired
at
fully
is
utilized
capacity
All
3)
one
least
product,
capacity
that
is
both periods.
in
utilized
fully
is
dedicated
the
in
at
one
least
of
two
the
periods.
4)
It
never
is
product/period
pairs
any
optimal
inventory
carry
to
which flexible capacity
in
product/period
is
which
out
both
of
used.
5)
In
all
dedicated capacity for that product will also be employed.
used,
also
is
It
possible
pair
in
make statements
to
capacity
flexible
relating
production
product
in
is
and
sales.
There
6)
which
sales
other
period.
7)
for
product are
that
sales
If
never slack capacity
is
for
Sales
capacity
flexible
in
for
is
at
least
at
a
as
great
as
a
they
period
are
in
the
in
product are constant, capacity dedicated to that
a
product will be fully utilized
8)
for
least
used
both periods.
in
one
of
the
product/period
pairs
must be greater than sales for
which
in
that
product
other period.
the
9)
the
If
optimal
solution
has
sales
of both
products equal
in
each period, then no inventory or flexible capacity will be used.
Combining these properties gives
Theorem
KSDP,
4.
If
1
it
optimal
is
to
the
following
useful
result.
purchase flexible capacity for the
then in the optimal solution either:
Zai = ^82= K^B. Za2=Zbi =
(Ya2 = Ka or Ybi = Kb
or
or Ib2 =
(1^1 =
(
Sai> Sa2
or Sbi <
0,
Y^i = Ka, Yb2= Kg,
or both),
both), and
Sb2 or both),
or
2ai=Zb2=0,Z;^=Zbi= KaB' Ya2= Ka, Ybi = Kg,
(Yai = Ka or Yb2 = Kb or both),
(Ia2 =
(
And,
or Ibi
Sai< Sa2
if
Za1=Za2=
it
is
Zbi =
=
or both), and
or Sbi > Sb2 or both).
optimal
to
purchase
Zb2=0-
29
no
flexible
capacity,
then
As
a
result,
optimal
all
belong
solutions
one
to
of
three
The two
Kab =0 and two with Kab > 0categories with Kab >0 are symmetric (See Figure 4.1.) because
either product could be called product A and, by Assumptions A 10
one
categories,
with
and All, either the odd or the even periods could be labelled with a
Figure 4.1
Optimal Patterns of Allocation of Flexible Capacity
Za2- Kab>
Zbi =
Period
Product
A
Product
B
X
Period 2
1
X
and
linear
expect
particular,
can
lead
use
flexible
inventories
demand."
just
the
i.e.
The
the
true
the
A
Product
B
KSDP
that
do
present
and/or
minimum
that
in
will
be
in
every
dedicated
the
at
least
sold
in
any
fori = A,B.
30
demand
always
hold.
appealing
period
capacity
might
and
In
heuristic
to
state,
coupled
meet
and
with
demand in excess of base
how to meet swing demand.
as much dedicated capacity
symbols,
k' > Min{s-j}
fact,
intuitively
does not specify
firm to purchase
used.
one
'properties'
"Use dedicated capacity
meet 'swing demand',
heuristic
several
following
is
with unitary inelastic
not,
solutions.
demand
capacity
to
tells
that
suboptimal
to
demand',
'base
be
to
shows
it
of a
demonstrates
costs
intuitively
as
Product
Period 2
1
denotes a product/period pair in which flexible capacity
The following example
It
Period
state
and
period.
In
This
heuristic
simplifies
demand
so
the
analysis
meeting
swing
demand.
problem
the
on
can focus
by
the
'subtracting
more
base
out'
problem of
difficult
Example
rA =
rB=
r^
1.0
Dai =5
Da2 =
=
1.1
Dbi=0
5
^Al = hA2 = 0.25
Db2 = 10
hBi = hB2
Rai = Ra2 = Rbi = Rb2 =
By Hbi = hB2 = «> it is meant
=
~
°°-
holding costs for product B
that
high the firm would never inventory product B.
= Rbi=Rb2='*' means
demand
always
is
problem reduces
Using
met.
to
ensure
KSDP.
the
the
following linear program:
the
to
of
properties
the
so
Rai = Ra2
Similarly.
high enough
stockout costs are
that
are
Max -Ka-Ke-1.1Kab-0.1Iai
s.t.
Ke+Kab^
Ka+ Kab-
Iai
Ka+Iai
5
^
10
^
with Ka, Kb, Kab. and
The
>
>0
5 and
heuristic
Kb ^
Ka =
is:
optimal
12.25,
5,
is
solution
if
nonnegative.
Iai
K^ ^
Min\
=
Ka
0,
minimum
It may
be
to
5,
to
for
the
its
given
a
to
some
of sales
optimal
demand
is
over
to
all
use
striking
state/period
flexible
known and
purchase no capacity dedicated
to
3
I
that
is
^
5
It
and Kg
The
15.
true
a cost of
parameter values,
gives
the
optimal
conclusions.
much dedicated
capacity as
pairs.
capacity
constant.
cost
Its
Ka
and Kab = ^0 ^^^^
not always optimal to purchase as
the
if
So,
=
0.
assume Ka
safe to
by
constrained
and Kab =
0, Iai
is
it
suboptimal solution.
1.375 or if hAi = hA2 > 0.8.
leads
rAB >
It
even
0.
Kg =
implies that
solution
completely different.
1)
2)
=
10, Iai
This example leads
is
S,j/
The optimal
Kb =
solution:
heuristic
the
0.
5
In
to
fact,
product.
produce
it
may
a
product
be optimal
and
capacity
Flexible
3)
inventory
can
1.375 no inventory or flexible capacity
is
used,
is
A
optimal to inventory 5 units of product
each even period and
decreasing
the
acquire
to
cost
of
complements.
be
but
if
<
r^^^
1.375,
from each odd period
10 units of flexible capacity.
can
capacity
flexible
increase
1-1'
if
unit,
no
longer
is
it
holding
increasing
Properties
KSDP
The
A5.
called
is
Quadratic-Known Seasonal Demand
the
strictly
cannot be guaranteed,
but
if
the
model parameters
set,
then
the
any continuous
is
Q-KSDP many
model
(There
parameters
continuous
are
condition
is
satisfied
satisfied,
solution
will
violate
solution
that
does
could
in
with
it
the
not
is
property,
violate
principle
the
be
probability
1,
in
and
any
capacity:
32
there
property.
violated
"knife-
certain
a
each property.)
for
according
open
any
the
by
these
the
an
optimal
always
an
optimal
Hence,
an
if
that
case
is
any
to
set,
although
optimal
solution
these
solution,
See Caulkins (1988) for
optimal
open
solutions
Furthermore, even
0.
always
not
can safely be assumed that they hold.
With
distributed
probability
of optimal
satisfy
over
for
0.
conditions
distribution
probability
conditions
properties
randomly
are
any
over
distribution
exactly
different
are
distribution
probability
additional properties
parameters
are linear,
uniqueness of the optimal
nonuniqueness occurs with probability
condition.
Q-KSDP
and so violate Assumption
convex,
that
edge"
flexible
in
revenue functions and linear holding
is
the
So,
Q-KSDP
the
to
this
unless
Similarly,
capacity.
investment
Since the holding costs for the
convex but not
For the
Thus,
increase above 0.8 per
flexible
less
to
Solutions
with quadratic
the
If
lead
The only consequence of
solution
hold
can
of Optimal
Problem (Q-KSDP).
are
purchase
to
to
smaller inventories.
and production costs
they
optimal
costs
capacity as well as
4.2
A
holding costs for product
it
optimal
inventory levels as well as investment in flexible capacity.
with r^B =
1^3 >
If
using
it
details.
flexible
The firm
(1)
in
not
will
which flexible capacity
one
If
(2)
both
or
inventory
carry
both
into
product/period
pairs
used.
is
dedicated
of
types
capacity
used
are
then
inventory of both products will not be carried into the same period.
capacity
If
(3)
not
is
fully
utilized
some product/period
in
no inventory of either product will be carried into
both
If
(4)
carried
types
into
possible
is
leads
to
some
very
long,
inventory
used,
are
only
pairs.
Q-KSDP. and
completely solve the
to
The derivation of
useful insights.
is
be
will
Q-KSDP
of the
It
capacity
period.
that
most one of the four product/period
at
Solution
4.3
dedicated
of
then
pair
described
briefly
here
solution
this
which
solution,
the
but
can
and
Zjj for
found
be
is
in
its
entirety in Caulkins (1988).
There are 15 decision variables:
1,2
and
for
r;
i
of which variables
are
positive
are
their
at
not
but
boundary
of
derivative
of the
objective
variables
equal
to
Solving
optimality.
order
conditions
expressions
strategy
properties
in
optimal.
It
decision
there
that
sufficient
Section
is
the
of these
free
condition
for
first
form
when
values
optimal
possible
conditions
expressions
for
its
for
derive
to
the
in
It
a
show
one can
4.1,
category.
optimal within
conditions,
setting
closed
yields
variables
variables'
the
comprised of the
of equations
free
on
that
over four million strategies, but using the
are
each of the strategies
within
the
all
necessary
upper
variables
be
so
strategy,
a
a
=
optimal.
is
In principle
for
of
gives
system
the
the
for
always
to
their
definition,
with respect to one
function
zero
By
at
bound cannot
upper
their
at
j
defined as a specification
is
upper bound.
corresponding
region
the
strategy
= A.B and
i
which are positive but not
zero,
bound, and which are
that
A
= A, B, and AB.
Yij,
Ijj,
is
only
that
necessary
and
strategy
in
a
decision
variables
to be
optimal
find
necessary
category
with
K^g >
category for a fixed value of Kab-
the
conditions
to
possible
also
sufficient
K^^ =
category with
can ever be
137
derived
from
and
to
Then using
the
first
be
the
order
one can compute the optimal value within the category as
33
a function of
categories
three
optimal
Comparing
K^^^^.
for
of
within
and
period,
optimal
values
No
one
these
of
intuition
the
overall
the
Sensitivity
Since
decision
variables
have
all
objective
itself
function.
is
particularly
is
the
interpretations
and
results
for
key
to
developing
Q-KSDP
the
gives
form
closed
strategies
all
expressions
terms
in
of
of any function
Y
the
for
the
problem
of the decision
be computed
rule:
dY*
aY*
dx
3x
partial
aY* 5Kab
derivatives
If the
^AB =
^nd there
(4.1)
3x
-.T^*
optimal solution
directly.
0'
for
the sensitivity
K^^^,
each
in
general.
in
for
by
cost
expressions
the
and
analysis
sensitivity
demand
with respect to changes in a parameter x can
by the chain
The
the
holding
conditions
variables
give
and
of
do
as
interpretations
to
solution
the
and
decision
able
Analysis
variables
parameters
interpretations,
economic
problem
The
be
to
conditions comparing
strength
K^.
strategy
a
demand
the
to
for
types:
relative
on
the
following
the
for
of
being
but
significant,
the
economic
the
obtaining
on
conditions
straightforward
of three
are
capacity
conditions
conditions
sufficient
category
its
dedicated
of
cost
parameters,
3Y
—
—
Z
is
and
in
3Y
no alternate solution with
is
can
be
computed
the category of solutions with
K^
dK*
>
0, then
dx
*
0.
yields
Kj^q
solution.
optimal
4.4
values
possible
all
The necessary and
the
optimal solutions within each of the
the
For solutions with
K^
>
*
'AB
(4.2)
ax
34
V^ denotes
where
denotes
the
second
the
derivative
partial
derivative,
partial
by differentiating the
obtained
_a_
av
(Ka^x),
a
x)
Rearranging
By
strategies
that
gives
can
aV (K^x).
aK^^x)
^^
be
=
0.
^^AB^'
^k^k^
optimal
be
V,,
x,
formula can
This
etc....
to
(4.2).'
computation,
direct
x|
aK^B
terms
respect
0.
V (Ka^x).
dK AB
ax
with
order condition
first
av(KA^xU)
=
dK AB
V
of
nonzero
(with
negative
strictly
is
for
probability),
so
all
(4.2)
aK vD
does
not
division
entail
bv
Furthermore,
zero.
sgn(
.
)
=
dx
sV
With
and
(4.1)
variable
to
straightforward
any
of
but
particular
strategy
the
derivatives
of
do
have
and
strategies
the
all
not
if
all,
of
that
increase
or
will
etc..)
same.
remain the
used
be
Similarly
to
for
'decreases'.)
and using
amount of
used,
insights.
can be zero for certain combinations of strategies, the
The most important of
inventory
the
economic
interesting
(Since most,
so they will be described.
quantities
term
on
general
yield
term 'increases' (increasing, greater than, more,
describe
were
however, are the same for
derivatives,
the
not
depend
they
a
is
derivatives
the
because
here
decision
parameters
model's
of
any
of
sensitivity
values
hence
and
used,
The signs of many of
interpretations,
The
reported
not
are
the
reduced
the
calculation.
computed
combinations
obtaining
(4.2),
capacity
flexible
into
a
not
the
in
product/period
pair
is
product/period
pair
and
a
for
the
which
in
using
complement
35
that
in
of carrying
cost
substitute
a
is
holding
always substitutes.
pair
Hence,
pairs.
results
increasing
is
product/period
and decreasing
product/period
are
capacity
flexible
these
flexible
inventory
flexible
carrying
to
cost
The optimal
carrying
of
capacity
out
is
of those
capacity
inventory
carrying
inventory
in
into
inventory
out
a
that
of
product/period
that
product/period
analysis
sensitivity
one
with
associated
product
are
one
with
associated
priori
which
to
allocated,
general
complementarity
the
reveals
how
affected
by
decision
changes
or
product,
^
=
the
sensitivity
for
variables
parameters
in
Y
For any function
associated with the other product.
variables
be
will
about
a
and flexible capacity.
of inventory
substitutability
The
made
be
known
not
is
it
capacity
flexible
pairs
cannot
statements
Since
pair.
of the decision
parameters x
all
dx
with
associated
by the second term of
entirely
a
product,
other
the
(4.1),
parameter associated with product
value of y^AB' then
amount
of
-1
since
product
absolute
value.
product
i's
dedicated
other
the
holding costs
in
affect
a
the
capacity
dedicated
to
product
may
other
period
but
generally has
increased
the
out
of
that
the
investment
direction
of the
amount no greater
in
Thus changes
in
.
amount
of
their
effect
capacity
on
the
other
capacity
flexible
Production
product.
decrease,
overall
but
firm
the
period.
demand
If
capacity
effect
for
product
a
it
will
and
used,
is
thus
of
less
other
the
production
will
inventory
less
of the
also inventory
less
of the
will
Increased
dedicated
on investment
36
demand and
in
demand
opposite effect on inventories.
in
optimal
the
which flexible capacity
in
more
As one would expect,
product
through
solution.
product out of that period, but
first
to
or
change
shows how changes
pair
purchase
increase
only
also
optimal
the
—
the
with
associated
(dKABl
= sgnl—
K[,
capacity.
product/period
to
increase.
product
analysis
optimal
is
will
it
optimal
the
affect
flexible
sensitivity
increases
sgnl
parameters
to
The
/d|K-+KABJl
Hence
amount of
it
parameter
a
opposite direction and by an
the
in
changing
if
Furthermore,
product.
other
—
K"
Hence,
.
does not influence the optimal
i
capacity dedicated to the
dK'1
^ 377
^ 0, if changing
value of
then
^^
changes the optimal value of K^b.
i
optimal
.
determined
is
not influence the optimal value of
will
it
so
in
to
the
flexible
in
the
other
It
also
leads
first
product,
capacity and
capacity dedicated
in
is
other product depends
the
to
on which strategy
in use.
If
in
which
Production
possible
that
production
of the
second
product
surprisingly,
On
of
out
than
out
decrease
of
inventory
If
costs
more
and
direction
of
overall
values.
carry
the
will
and
pair
purchase
less
capacity
became
product
may
Finally,
the
all
demand
down.
Inventory-
product,
either
so
total
if
more
costs
a
pair.
product,
first
the
parameter
inventory
used,
is
will,
It
on
carry
to
capacity
flexible
such
into
depend
levels
the
to
The
product.
other
the
to
dedicated
capacity
it
purchase more flexible
will
into
firm
the
a
will
however, increase
its
same two periods.
the
price of one of the dedicated capacities increases the firm
capacity.
are
magnitude
product.
first
for
firm
the
other product between
flexible
other
the
dedicated
which
in
inventory
the
of
in
inventory into a product/period pair in
production
Not surprisingly,
inventory of the
If
on
effects
capacity,
less
product/period
that
smaller
be
will
affected
used,
is
capacity
less
product/period
not
carry
to
which flexible capacity
capacity
inventory
is
of
Not
decrease.
levels
it
in
overall
in
decrease.
will
out
less
increase
that
period
other
the
decrease
the
increase
an
to
is
it
inventory more of the other product
will
it
the
in
rise
will
and
increase,
offset
production
inventory
will
but
period,
that
giving
overall
other hand,
the
the
and
than
product
that
cenainly possible that both production of
is
it
firm
the
more
will
of
may
product
other
the
product,
first
Of course
production.
pair.
of
production
used,
is
increase
its
inventory out of a product/period pair
carry
to
capacity
flexible
decrease.
the
more
costs
it
consistent
increases,
of
both
kinds
Production
more
of
expensive
increase
derivatives
with
profits
or
the
whose
product
more
dedicated
production
decrease;
will
and
of
the
decrease.
of
one's
capacity
dedicated
of
the
optimal
intuition
increase.
37
objective
about
When
the
costs
function
model.
go up.
value
When
profits
go
Conclusions and Discussion
5
The
advent
manufacturing
flexible
economic
the
assessing
of
value
of
capacity
conditions so
extends
value
gives
the
case
when
assessing
firm can
the
the
hold
and
building insights.
intuition
FMS
to
for
This paper
on the complete solution for a special case,
reports
some
The
T+1
FMS's
of product-flexible
inventory,
explore both in one model.
to
Freund (1986, 1987) model
and
Fine
the
useful
is
systems
both ways firms can respond to varying market
are
it
manufacturing
flexible
made
has
Holding inventory and investing in
(FMS's) an active research topic.
flexible
technology
investment decision problem was
formulated as a
first
programming
problem called the General Stochastic Demand Problem (GSDP) that
explicitly allows the firm to hold inventory.
Few analytic results
investment
period
GSDP
can be obtained for the
uncertainty
market
about
Section
dynamic
production
some
unless
imposed on the
structure is
conditions.
considers
3
and
GSDP when
the
variability
the
in
market
known when production and inventory
decisions are made but unknown at the time the firm makes its
With this structure it is possible to show a
investment decision.
conditions
and
periodic
is
unique optimal solution exists; give explicit formulas for the optimal
Karush-Kuhn-Tucker
conditions for
optimal
value
characterize
for
the
to
it
a
is
the
special
multipliers;
and
necessary
state
sufficient
be optimal to purchase flexible capacity; show the
and
well-behaved function of the capacity costs;
optimal
solution
as
a
function
case of quadratic revenue and
of
capacity
the
costs
holding cost functions
and linear production costs.
4 reports
Section
market conditions
its
investment
stated
for
is
drawn from
functions and
the
the
periodic
decision.
general
on
and known
Many
revenue
case
special
and
at
properties
cost
when
time
the
of
the
the
optimal
functions
as
variability
are
firm
in
makes
solutions
are
conclusions
complete solution for the case of quadratic revenue
linear
production and holding costs.
38
Two
can
capacity
determining
allowed
be
intuition
can be misleading.
is
for
and
results,
such
numerical
those
as
to
from
results
special
expected
to
capacity
not
Although
2)
demand
is
produce
a
demand
is
not vary
3)
demand
of
the
of
modellers
and
Section
4
problems,
those
the
solution
is
a
but
a
that
paper will
Some
practitioners.
are
on
hoped
is
this
in
The
rule.
depends
the
it
are
practice,
In
the
Likewise,
and
long,
closed form
solutions
and the examples
case
is
unguided
that
that
be
guarantee
that
3,
is
it
when inventories
probably
parameters.
consequence of the
can
following
the
is
to
investing
be
of
holding
flexible
in
market variability, the two
They may be complements,
not possible to
cost
and
inventory
how changes
or
make
substitutes,
general statements about
inventory
the
affect
value
the cost of flexible capacity
in
of
affect
levels.
may be optimal to purchase flexible capacity even
In fact, it can be optimal
known with certainty.
exclusively
product
known
with
with
flexible
and
demand
certainty
even
capacity
for
if
if
to
all
product does
that
over time.
Sometimes
into
two
uncertainty between
first
it
the
in
inventory
It
Section
in
substitutes.
flexible capacity
optimal
the
models.
numerical
ways firms respond
both
how changes
from
holding
Hence
or neither.
considers
importantly,
capacity
will
that
are
showed
realistic
and
more generally.
hold
always
solutions
special
of
more
flexible
inventories
only
seems unlikely
it
flexible
for
examples
the
are
of
intuition
the
structure
1)
are
of this
analysis
contribute
4
example
the
Hence
inferences
well-behaved function
the
and
obtained
draw
to
value
the
be
will
ability
when
more
and
that
certainty
of
Section
in
are
4
investment
model
complex,
itself
intuition
world
a
flexibility,
solution
allowed
Section
Nevertheless, the derivation of the solution
deterministic.
expressions
in
FMS
a
The
product
of
benefits
of
difficult.
is
even
useful
value
the
from
conclusions
overall
it
is
useful
components:
periods.
to
decompose
uncertainty
Flexible
kind of uncertainty; inventory
is
39
capacity
between
is
about
uncertainty
products
most useful
most useful for
the
for
second.
and
the
4)
demand
'base demand',
met
with
use dedicated
to
knows
firm
the
Thus
heuristics
dedicated
capacity,
and period.
state
be
always optimal
not
is
It
will
for certain
that
'subtract
leaving
capacity
exist
in
every
base demand, to
out'
smaller
a
meet
to
and
hopefully
simpler problem can lead to suboptimal solutions.
The ultimate objective of research
of FMS's
investment
models
develop tools practitioners can use to
to
is
the
process, so
it
be done
the
at
require
will
literature
them on
testing
Developing
making.
decision
in
economic evaluation
the
in
world problems.
real
important that
is
a
level of
practical
the
including
This will be a long and difficult
begin now, but there
it
from
tools
of work,
deal
great
technology
assist
is
also
more
to
academic research.
This paper was based on the Fine and Freund (1986, 1987) model
quantifying
for
number of models
the
in
product
than
other
value
the
FMS's.
product-flexible
of
flexibility.
focus
that
literature
is
It
likely
There
are
a
on benefits of FMS's
extending
that
them
to
consider inventory effects would yield useful insights.
There
are
and/or invest in
cycle considerations, and economies
life
have
between
models
and
even
inventories
and
flexible
include
that
statements
focuses
all
about the
exclusively
other
the
different
quite
three
effects
unlikely
of
to
the
on
Hence
capacity.
nature
overall
hold
inventory
These factors can
of scale.
opposing
are
firms
market conditions, product
varying
capacity:
flexible
why
reasons
three
least
at
yield
the
relationship
comprehensive
simple,
relationship.
qualitative
This
paper
on the effects of varying market conditions, but
two deserve study.
Further work could center on these related problems, but there
how holding inventory
affects the value of product-flexible FMS's when market conditions
vary.
The analysis in this paper rests on many assumptions and
simplifications.
It
would be useful to know what results presented
here continue to hold when these assumptions and simplifications
is
also
are
certainly
to
be
learned
about
relaxed.
Direct
may
more
not
be
generalizations
the
of
the
model
most productive avenue
40
presented
for
further
here,
however,
research.
The
how holding
problem of determining
no
flexibility
with
conditions
is
possible
because
to
analyze.
considered
it
Periodic
conditions.
structure
particular
too general
inventory
conditions
are
to
affects
variability
the
The analysis
in
value
the
this
market
in
thesis
varying
periodically
complex enough
to
of
was
market
bring
out
the
dynamics of the interaction between inventory and flexible capacity,
but
simple
explore
be
enough
other
interesting
analysis.
facilitate
of market
types
but
to
simple
variability
enough
to
simpler
above,
that
constituents,
variability
between
overall
such
as
that
variability
41
are
be
useful
to
complex enough
to
may
the
results
the
idea
decomposed
between periods
into
to
can
variability
products.
would
analyze and compare
The key
obtained with those presented here.
mentioned
It
this
be
be
and
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&
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s
s
Appendix: Proof of Theorem 3.2
Proof:
KKT
conditions
except perhaps mi for
be
complete
By
AB
= A, B, and
shown
is
it
V
satisfies
so
the
direct
substitution
they
these
and VI.
So, the proof
multipliers
three
Let X be any
set
—
shown
be
will
it
V.
satisfies
are
of optimal
_S
X
that
variables
all
are nonnegative.
that
multipliers.
First
the
obvious that
is
it
in
determined
uniquely
is
and
IV,
-
I
i
if
and X
nonnegative
KKT
E
3.1,
defined
multipliers
above are uniquely specified.
multipliers
will
of
collection
the
By Theorem
theorem.
satisfy
denote
Let X
_S
Sij-qii
=
s
s
_s
So since the optimal solution is unique, by III,
Y Ps\ ^i2~hii- Rii/.
~s
~s
—s
^s
_S
—S
_s
_S
_s _s
Likewise, 5i2- qi2= 5i2 - q,2
Since 5ij q^ = 0,
5.i-qii = Sii - qn~S
^S
_s
_s
>
and
Similarly, the formulas in the
J
q^ > q^ for all i,j, and s.
5ij
5,
_s _s — s ~s ~s __s
.
•
Theorem
KKT
and
Transposing
condition
~S
s
a,j=a,j +
terms,
~s
~s _s
aij>aij.
Similarly,
multipliers,
imply
_s
/
_„S
(
5..-
_s
- Sy+
Because X
.
is
Hence, since a >
V.
5^-
Sij=
a^-
—
~s
-s
\
5.J
a,j-
Since
Sjj.
5,j- s^.
—
s
a^
Sij=
0,
~
Yj^Yj
satisfies
it
I
an
optimal
y >
a,
set
of
and q >
"i,
KKT
q,
X
also satisfies V.
—*
~'
Next
will be
it
= Oifi"J<rij. So
S
Then
and
since
KKT
shown
7jjY-j
_S
if
satisfies VI.
s
Since X satisfies VI, SyYjj
=0.
Suppose
l\>'s;y
Also, from the formulas in the
Theorem
7jj>rjj implies
Y-j
s
a^ Sij=0,
condition
=
X
I,
0^,
7jj
=0.
=
7'j
~
+ (5ij-5j-(aij- a-J
•
So
if rij>s'ij,
it
must
~S
_s
be that
Since X is an optimal set of multipliers, by V
5,, > 5;;.
~s
^%
—S — s ~s
S
S
S
implies
Furthermore,
since
that
>
I,j=Y,j+Zij.
q^,
5ij
5ij- qij= 5ij
~s
—
_S
_s
s
and thus
By VI this implies lij =
Sjj > 5ij implies that q^j > qi,>0.
Y-j
=0. Hence
s"ij>7ij
implies
yJj
=
0.
46
So since
SjjYJ,
=
0,
s^,jYJ,
=
0.
Next
"ij
implies that
~s
V^
t'y
=
y'j
~S
_s
>
q,,
llj '
q,
MlJ-
—
+
zJj
Next
,
>
Yj^Yj.
~
..
.
~s
_s
u,j>Uij.
>
since
6, - q,
an optimal set of multipliers,
~»
Likewise, 5,j >
implies that
is
s
lij=0.
=0. Hence u,j>u,j implies Z-j=0,so
be shown that mAKA=0. If mAKA>0,
= 0.
so Z-j
UjjZij
will
then 01^ >
it
_
K^
and
Then
S.-q,-
Since
5,j>5,j.
Since X
.
implies
qij>qij^O
Suppose
that UijZij=0.
and
= "u+l5ij-5j-[Yj-Yj
this
I-j
shown
will be
it
Ka>
0.
s
mA=
implies
_s
2
So by IV, rA= XX°^Aj. and thus
.
S=l J=l
"^A =
2-2^\"aj~ *^aJ^
This implies there
.
some
exists
state/period
S=l J=l
pair
~s
_s
a,j>a,j.
such that
(s,j)
~s
implies
_S
5ij>5jj,
~s
implies
S
V
which by
implies
S
_s
which
q,j>qij.
S
Z^>
implies
~s
Finally,
m^
~*
shown
=rA-SS«A. ^
mg
>
m^
and
>
So X
0.
X
Furthermore,
K
>
K
multipliers.
by
IV
>
implies
and
the
XS"aj
a set of optimal
unique
the
is
and
So X
m^
=
nonnegative.
are
m^
VI.
satisfies
^
Similarly,
0.
S=l J=l
Suppose X and X are
0.
Hence
is
qjj
r^ -
S=lj=l
to
s
lij=0.
m^,mB,
that
required
are
steps
""
—s
s
and q^ lij=0,
q,j
will be
it
Since aij>a,j,
_s
>
q|j
So
Contradiction.
Iaj=0.
Similarly mBKB = 0.
Several more
'^aKa=0show m^K^ = 0, but the reasoning is similar.
Also, since
.
s
VI
by
Ya,= Ka>0.
Yaj+
Iaj=
~S
_s
Also a,j>aij
~S
_s
Buta,j>a,j also
s
Then by V,
tw^o
m^ = mg
formulas
in
set
KKT
multipliers.
KKT
of optimal
distinct
vectors
= m^B =
the
of
m^
=
Theorem,
multipliers
if
KKT
optimal
= mB = rn^e
XX ^aj
=
""a
•
=
s=lj=l
S
2
^^5
^^ s
XS^aj-
,^ s
s
"^
s
So since a,j>a,j, a,j=a,j
for all
i,j,
and
s.
Similarly,
Y = Y.
s=lj=l
Then
unless
by
I,
~S
_s
5„ >
5,>0
5
-j
+
F-j
= 5
_S
and
,j
+
T.j
.
Since
5,j>5,j,
5,j+ s,j=
~
~S
s„> s,>0.
47
But since X
6,j
VI,
s,j
~S
s,,>s,,
-5
satisfies
+
—
5
implies
Yy =
implies
ujj
0.
Similarly,
> Uij>0 which
Assumption A6,
Y* =
smce
implies
0<Ki
—
Y=Y,
~S
_S
5ij+ Uij= 5,j
Zij=0.
Yij= Z-j=
implies
~S
_S
lij=0.
Then by
_S
__s
+ u^, so 5^ >
implies
I,
I-j
=
.
YPsR-j(O)-
5ij
By
cJj(O)
= aii-Sii-Sji- But Sij>0 implies 0^=0, so r[j(0)- cjj(0)<
which
Thus
And since
contradicts Assumption A8.
5 = 6, s = s, andu = u
_
~
~
~s
~s
_
_s _s
q = q. Thus X = X, contradicting the hypothesis that
5ij+ Sij= 5ij + Sij,
•
they are distinct.
So
if
K
>
0,
the
formulas above are unique.
k^hl
U32
48
optimal
multipliers
given by the
Date Due
Lib-26-67
MiT
3
'=1060
MBRARIF<;
niiPi
J
00Sb?M33
5