Economic Analysis of Product-Flexible
Manufacturing System Investment
Decisions
by
Charles H. Fine
Robert M. Freund
Working Paper #1757-86
Sloan School of Management
Massachusetts Institute of Technology
02139
Cambridge, Massachusetts
Comments are welcome.
March
1986
Economic Analysis of Product-Flexible
Manufacturing System Investment
Decisions
by
Charles H. Fine
Robert M. Freund
ABSTRACT
This paper presents a model of the tradeoffs involved in
investing in flexible manufacturing capacity.
Our model assumes
that the firm must make its investment decision in manufacturing
capacity before the resolution of uncertainty in product demand.
Flexible capacity provides to the firm the ability to be responsive
to a wide variety of future demand outcomes.
This benefit is
weighed against the increased cost of flexible manufacturing
capacity vis a vis dedicated or nonflexible capacity.
We provide a general formulation of the two-product version of
this problem where the firm must choose a portfolio of flexible and
nonflexible capacity before receiving final demand information.
Under the assumption that demand curves are linear, our model is a
two-stage convex quadratic program.
Our results characterize the
optimal profit function and the optimal investment policies.
Furthermore, we present general sensitivity analysis results that
show how changes in the acquisition costs of flexible and
nonflexible capacity affect the optimal investment decisions.
Finally, we present a numerical example where we compute the optimal
investment decisions and the value of flexibility in the context of
an automibile engine plant investment decision.
Economic Analysis of Product-Flexible
Manufacturing System Investment
Decisions
by
Charles H. Fine
Robert M. Freund
Introduction
Advances in microprocessor-based manufacturing technologies
have hastened the development of automated manufacturing systems
Compared with the
that are being noted for their flexibility.
less automated systems
they are designed to replace,
flexible
manufacturing sytems often cost more to acquire and install,
yield lower variable costs,
and greater flexibility.
higher product conformance
but
(quality),
Management analysts have a great deal
of knowledge and experience in how to evaluate investments that
reduce operating costs,
experience
less
a moderate amount of knowledge and
in evaluating cost
and quality tradeoffs,
and much
knowledge and experience in evaluating investments
that
enhance flexibility.
The model
in this paper
analyzing investments
of
the economics of
flexibility has
The subject
been of interest to
(see, e.g.,
in Jones and Ostroy
significant interest
to the knowledge base on
in manufacturing flexibility.
economists for a long time
many references
contributes
Stigler
[19841),
[1939],
and
the
but has become of
in the OR/MS community only recently,
following the increasing viability of flexible, computercontrolled manufacturing systems.
a
large amount of work in
and Suri
[1984],
Adler
This new interest has spurred
a very short time.
1985],
Kulatilaka
1
(See, e.g.,
1985]).
Stecke
11
The fundamental
decision problem
timing of
structure of
the flexibility investment
is quite simple.
The central
investment and production decisions
of uncertainty.
Our model
assumes that
plant and equipment before they receive
on product
demand.
Thus,
issues are the
and the resolution
firms must invest in
their final
information
firms will prefer plant and equipment
with manufacturing flexibility so that they can
set production
optimally once the final demand information becomes
This desired
flexibility can be
but only at a cost.
obtained at the
Herein lies the tension
the real-life decision
in
available.
investment stage,
the model
(and in
problem).
We provide a general formulation of the two-product version
of this problem where
the firm must choose a portfolio of
flexible and nonflexible
information.
Our results
function and the optimal
capacity before receiving final demand
characterize the optimal profit
investment policies.
Furthermore,
provide a sensitivity analysis to show how changes
acquisition costs
optimal
we
in the
of flexible an.d nonflexible capacity affect the
investment decisions.
Our model
focuses on investments in product-flexible
manufacturing systems
(PFMS).
By a product-flexible
manufacturing system, we mean one that
can produce a number of
different products with very low changeover costs and
times.
This definition of flexibility is consistent with Mandelbaum's
(1978]
and Buzacott's
[1984]
production flexibility.
for a review of the
[1982]
state
flexibility and Browne's
(See also Adler
often confusing,
classifications and definitions).
2
[1985] Section 4
non-standardized flexibility
In this paper, we
concentrate on investments
Even in a world of certainty,
uncertainty.
in PFMS under
product-flexibility
is valuable, because equipment downtime for changeovers
affected favorably by the use of flexible equipment.
is
Tradeoffs
of this sort are appropriately analyzed by comparing total costs
of system operation after optimizing the scheduling rules for
both flexible and nonflexible systems.
paper at a more aggregate level,
complexities for PFMS.
In order
to focus our
we assume away the scheduling
We think that these
scheduling issues are
very important, but this modelling choice allows us to
concentrate on the effects of
the
uncertainty in product demand and
(short-term) irreversibility of investment decisions on the
optimal
In
mix of flexible and nonflexible
the next section we present
derive some basic properties
properties of
examine
of the model.
investment levels
Section 2 presents
In Section 3 we
for flexible and
Section 4 presents a numerical example where we
optimal investment decisions and the value
flexibility in an automobile engine plant
Section 5
formulation and
technologies and provide a sensitivity analysis for
these resul ts.
compute the
our model
the optimal value function.
the optimal
nonflexible
manufacturing capacity.
of
investment setting.
contains a discussion of extensions and
implications of
the model.
1.
PFMS Model Formulation and
Basic Properties of the Model.
We assume that the firm can produce two different products,
A and B.
Before producing,
investment decisions.
Three
the firm must make capacity
types of capacity are available:
dedicated A-capacity, dedicated B-capacity, and flexible AB-
3
III
capacity.
We denote by KA, KB,
and KAB,
the amounts of each of
these types of capacity invested in by the firm.
purchase costs
for these capacity types are
respectively.
Before making production decisions, but after
making capacity investment decisions,
rA,
The per unit
rB, and rAB,
the firm observes a random
variable that provides information about the state of
for market demand for products A and B.
the world
We assume there are k
states of the world and the probability that state i occurs is
Pi,
where
Pi > 0
for all
i and
£
Pi
=
1.
After
observing state
i
i, the firm chooses its
production levels XAi and XBi of products
A and B,
capacity constraints
subject
to the
imposed by the
earlier investment decisions.
The
state
firm faces a linear demand curve for each product.
i, the price consumers
ei-fiXAi.
In
pay for each unit of product A is
Similarly, the demand curve for product B is gi-hiXBi.
For simplicity, we assume there
is only one period in which the
firm makes production decisions.
Finally, we assume that the
variable cost of production of a unit of product A
(or product B)
is the same whether a unit is produced with flexible or
nonflexible capacity.
Thus, without loss of generality, we
assume that the variable costs
implications
of relaxing
of production are zero.
this and
other
The
assumptions are
discussed in section 5 of the paper.
With this notation, we can formulate the PFMS
decision problem as PFM1:
4
investment
k
-rAKA-rBKB-rABKAB +
maximize
Pi(XAi(ei-fiXAi)
E
+ XBi(gi-hiXBi)
)
i=1
KA,KB,KAB
XAi,XBi,i=,
...
subject
,k
XAi-KA-KAB
< 0
i=l ... ,k
()
(6i)
XBi-KB-KAB
<
0
i=l,...,k
(ii)
(ei)
XAi+XBi-KA-KB-KAB < 0
i=l,...,k
(iii)
(Xi)
to:
0
XAi,XBi
KA
O,
>
KB > O,
i=i,
...
,k
> 0
KAB
(qi,ri)
(iv)
(v)
(m,n,p)
The notation for the Lagrange multipliers for each of the
inequalities
(i)-(v) is given to the far right of
each line.
We
make the following assumptions:
Al.
ei
> 0,
gi > 0,
i.e.,
the
intercept of each demand curve
is
positive.
A2.
fi,
hi > 0,
i.e.,
each product's demand curve is
strictly
downward sloping.
A3.
rA > 0,
rB > 0,
rAB > 0,
i.e.,
there are positive purchase
costs for all capacities.
A4.
rAB > rA, rAB > r B ,
i.e.
the per unit cost
of flexible capacity
strictly exceeds
the per unit cost of nonflexible capacity for
either product.
(Note that
if rAB <
rA and rAB
PFMS problem simplifies significantly because no
rB, then the
firm would
ever purchase any nonflexible capacity.)
A5.
rAB <
rA + rB ,
strictly less
i.e.
the per unit cost of flexible capacity is
than the per unit cost
capacity for both products.
obviously
of purchasing nonflexible
(Otherwise,
never economical.
flexible capacity is
When assumption A5
is violated,
the PFMS problem splits into separate investment decision
problems
solution,
for each
product, and
see Freund and Fine
5
each problem has a closed form
[1986].)
III
di
ci
= Pigi,
= Pihi,
are equivalent to ai,
ci,
in PFM1.
di > 0,
The Karush-Kuhn-Tucker
program.
XBi,
i=l,...,k,
KA, KB,
=
iei,
ai
= Pi
f i
,
Assumptions Al and A2 then
i=l,...,k.
is a convex quadratic
assumptions A1-A5 the PFMS problem
Under
XAi,
i=l,...,k
bi,
substitute bi
we will
For notational convenience,
(K-K-T) conditions ensure that
KAB constitute an optimal
solution
to
the PFMS problem
if and only if there exists nonnegative multipliers
m, n, p.
Xi,
Si,
conditions
ei,
qi,
ri,
i=l,...,k,
for which the following
hold:
bi
-
2
di
-
2ciXBi
aiXAi
=
i
= ei
+
...
,k
-
qi
i=
+ Xi -
ri
i=l,...,k
Xi
k
rA
E
=
(Si
+ Xi)
+ m
(ei
+ Xi)
+ n
(Si
+ ei
+ Xi)
i=1
k
rB
E
=
i=1
k
E
rAB =
i=1
+ P
together with the complementary slackness conditions:
6 i(XAi
-
KA -
KAB)
= O
ei(XBi -
KA -
KAB)
= O
Xi(XAi + XBi -
and qiXAi
= O,
riXBi
KA -
= 0,
KB -
KAB)
i=l,..,k,
6
=
O
mKA = O,
nKB =
, PKAB = 0.
Theorem
(Existence and Uniqueness of Solutions)
1.
the PFMS problem has a unique optimal solution
assumptions A1-A5,
XAi, XBi,
PROOF:
KAB,
1:
PROOF:
is straightforward and
Under assumptions A1-A5,
and XAi,
of the
(see,
i=l,...,k, KA, KB, KAB.
The proof
Corollary
Under
XBi,
From the theory of
de Panne
piecewise-linear upper
single valued, and so
continuous.
rB,
optimal
values KA,
KB,
functions
and rAB.
parametric quadratic programming
1975]),
the set of optimal
solutions
is a
semi-continuous mapping of the linear
in the objective function.
coefficients
the
in the Appendix.
i=l,...,k are piecewise-linear continuous
capacity cost data rA,
e.g., Van
is presented
is a piecewise
By Theorem 1, this mapping is
linear function and is
[X]
Assumptions A4 and A5 play a critical role in guaranteeing the
uniqueness of
The necessity of
the solution in the PFMS problem.
is illustrated
these assumptions
in the following
nonflexible capacity are rA = rB = 10,
example.
Costs of
and there are two future
states, with data:
ai
bi
c
i
di
1
1/2
60
1/2
30
2
1/2
30
1/2
60
State
The table below shows
different values of
function of
i
optimal
rAB.
solutions to the PFMS
Figure 1 illustrates
rAB-
7
problem for three
the optimal
KAB as a
III
Solution
AR
EA
#1
10
0
0
#2
10
25
#3
15
#4
#5
EA EAf
au
A
B1
80
55
25
25
55
0
0
5
0
0
5
25
30
55
25
25
55
0
0
5
0
0
5
27.5
27.5
25
52.5
27.5
27.5
52.5
5
0
2.5
0
5
2.5
20
30
30
20
50
30
30
50
10
0
0
0
10
0
20
50
50
0
50
30
30
50
10
0
0
0
10
0
Upon setting q
= q2
= rl
= r2 =
,
m =
n = p = 0,
it
m
2
is easily
verified that the solutions above satisfy the K-K-T conditions, and
so are optimal.
A4.
is
Solutions #1
When rAB =
10,
rAB = r A
=
and #2 are each optimal
indifferent between building a mix of
capacity (solution #2)
Solutions #4
violating assumption
in this state.
The firm
flexible and nonflexible
and building all flexible capacity (solution
When rAB = 20, rAB =
#1).
r,
rA + r,
violating assumption A5.
and #5 are each optimal.
In this case,
the firm
is
indifferent between building a mix of flexible and nonflexible
capcity
(solution
4),
and building all nonflexible
capacity (solution #5).
Figure
1 shows
as a function of rAB in this example.
of KAB
= rB
values of KAB
Note that the optimal values
is a piecewise linear point-to-set mapping, and is a
continuous function over
rA
the optimal
< rAB
the range
10
< rAB < 20,
i.e.,
< rA + rB.
We now turn our attention to sensitivity analysis and
comparative statics
for the PFMS
problem.
facilitated by transforming the problem.
levels XAi,
XBi
= YBi
XBi,
+ ZBi,
Our analysis
For given production
these quantities can be split
where YAi
is
is
into XAi = YAi + ZAi,
that portion of total A production
being produced with nonflexible A-capacity, and
8
ZAi
is that portion
JI..
KAB
90
80 -
70 -
60 -
50-
40 -
30 -
20 -
10 -
-
I
0
5
Figure 1.
10
15
Optimal values of KAB as a function of rAB
9
S
20
rAB
III
of A production being produced with flexible AB-capacity
similarly for B production).
(and
With this change of variables,
the
PFMS problem now becomes
PFM2:
k
maximize KA,
KB,
YAi,
rB KB -
rAB KAB +
-
[ai
(YAi+ZAi) 2
[ci
(Bi+ZBi)
+ bi
i=l
k
KAB
ZAi,
i=l, ....
rA KA -
YBi,
ZBi
k
+
subject
to:
ZAi
YAi
-
KA <
YBi
-
KB
0
KAB
0
YAi
0
i=l
YBi
0
i =1 ,....
ZAi
0
i =1
ZBi
0
i =1 ,...,k
KAB
0
+ ZBi
KA,
i
-
KB,
2
YBi+ZBi)]
(i)
((ai)
, ... ,k
(ii)
(si)
,... k
(iii)
(Yi)
,...,k
(iv)
(si)
(v)
(ti)
(vi)
(ui)
(vii)
(vi)
(viii
(m,n,p)
i=l,.....k
i=l
+ di=
( YAi+ZAi) I
k
... k
It is a straightforward exercise to verify that PFM2 is equivalent
to PFM1, with the transformations:
XAi
= YAi + ZAi
Xbi
= YBi + ZBi
and
KA,
0)
- KB,
0)
YAi
= min
(XAi,
KA),
ZAi
= max (XAi -
YBi
=
(XBi, KB),
ZBi
= max (XBi
min
i=l,...k.
Theorem
1 asserts
KA, KB,
KAB, and
determined
that under assumptions A1-A5,
(YAi + ZAi),
(YBi + ZBi),
in an optimal solution.
10
the quantities
i=l,...,k are uniquely
In
the program PFM2,
decoupled and appear
prices a,
Bi,
the strategic variables KA,
in separate constraints.
and Yi are
easier to
value of an extra unit of A-capacity
we would
expect
and KAB are
Furthermore,
interpret.
marginal
KB,
the shadow
For example, ai
in state
i.
is
the
Therefore,
k
that at an optimal
ai =
solution,
rA, equating
i=1
marginal
value and marginal cost
k
of A-capacity.
Similarly,
we would
k
B
Si
expect
=
Yi
rB and
i=l
= rAB.
Indeed,
the optimality conditions
i=l
for the problem bear out this intuition.
PFM2 ensure that a feasible
i=l,...,k,
is an optimal
solution KA, KB,
solution to PFM2
exist nonnegative multipliers
i=l,...,k,
for which the
The K-K-T conditions for
m,
n, p,
KAB,
if and
i, Bi,
following conditions
YAi,
only if
Yi,
si,
hold:
(I)
bi
-
2ai
(YAi + ZAi)
=
ai
-
Si
i=1, . . .
k
(II)
bi
-
2a
i
(YAi
+
=
Yi
-
Ui
i=1, . . .
k
(III)
di
-
2ci
(YBi
+ ZBi)
=
Si
-
ti
i=1' . . . k
(IV)
di
-
2c
(YBi
+
=
-i
Vi
i=1, . . . k
(V)
rA =
i
ZAi )
ZBi )
k
E
+
i=1
(VI)
rB
(VII)
rAB =
=
k
E
i=1
k
E
Y
+
P
-=1
11
ZAi,
YBi,
ZBi,
there
ti,
ui,
vi,
III
i (KA -
Ai)
= 0
i
(KB -
Bi)
= 0
Yi
(KAB -
ZAi
-
(VIII)
(IX)
Si YAi
m KA =
=
O,
ZBi) =
= 0,
ti YBi
, n KB = 0,
Note that si,
t i,
u i,
0
i
ZAi = 0,
v
= 0,
i
i
i=l,...,k,
p KAB = 0.
vi, m, n, and p represent shadow prices on
the nonnegativity conditions
(iv) -
(viii) of the problem PFM2.
If
production with each capacity is positive in each state, then by
the complementary slackness conditions
= p = O,
i=l,...,k.
(IX),
Si
= ti
= ui
= Vi
= m = n
Condition (V) states that the sum of the marginal
contributions of nonflexible capacity for product A must be less than
or equal to the cost of nonflexible A-capacity, with equality holding
when A-capacity is purchased,
(by (IX)).
Conditions (VI) and (VII)
are interpreted in similar fashion for nonflexible B-capacity, and for
flexible AB-capacity.
The PFMS problem has a natural interpretation as a two-stage
program, where the capacity variables KA, KB, KAB are the first-stage
variables, and the production variables YAi,
second-stage variables.
and KAB,
ZAi,
YBi, ZBi are the
Given optimal values KA, KB, KAB for KA, KB,
in the PFMS problem, the k independent second-stage programs
P(i) are:
12
maximize
YAi,
ZAi,
YBi,
ZBi
-
ai
(YAi + ZAi)2
+ bi
(YAi + ZAi)
-
ci
(YBi + ZBi)
2
+ di
(YBi
subject to:
YAi
KA
(ai)
YBi
KB
(8i)
KAB
(Yi)
ZAi + ZBi
An optimal solution YAi,
+ ZBi)
YAi
> 0
(si)
YBi
> 0
(t
i
)
ZAi
> 0
(u
i
)
ZBi
0
ZAi,
(vi)
YBi,
ZBi to P(i),
i=l,...,k,
together
with KA, KB, KAB will constitute an optimal solution to the PFMS
problem, and under assumptions A1-A5, the total production quantity
(YAi + ZA1)
in program P(i) will be unique.
For notational convenience, let K = (KA, KBg,
vector of optimal capacity values.
By K >
KAB) represent the
we mean KA > 0,
KB > 0,
and KAB >
0.
Lemma 1.
Let K be optimal capacity levels for the PFMS problem
satisfying assumptions A1-A5,
solutions to P(i),
i=l,...,k.
and let YAi,
ZAI, YBi,
ZBi,
be optimal
Then the set of multipliers defined in
(1) below is an optimal set of K-K-T multipliers for the PFMS
problem.
Furthermore,
if K > 0,
then these K-K-T multipliers are
the unique K-K-T multipliers for the PFMS problem.
13
ill
=
[bi
-
ai
(YAi
+ ZAi)] +
,
Si =
[bi
-
ai
(YAi
+ ZAi)-
,
Si =
[di
-
ci
(YBi + ZBi)]
ti
[di
-
ci
(YBi
i
=
Yi = max (i,
Ui
= Yi
Vi =
i
+
+ ZBi)]-
Si)
-
i
+
-
i
+ ti
i
(1)
,
i=...,k.
k
= rA -
E
~i
i=1
k
n =
rB -
si
E
i=l
k
p = rAB -
£
Yi
i=1
PROOF:
The quantities YAi,
to P(i)
if and only if they form a solution to the PFMS problem
together
with the optimal
Theorem 1,
are
the quantities
ZAi,
YBi,
ZBi are an optimal
capacity levels KA, KB,
(YAi + ZAi)
= XAi and
YAi + ZAi and YBi
+ ZBi,
i=l,...,k.
PFMS problem.
conditions
m,
n,
(IX)
satisfy (I)
-
It thus remains
(VII)
to
defined in
By
= XBi
the
the formulas as functions of
All multipliers
nonnegative with the possible exception of m,
these multipliers
and KAB.
(YBi + ZBi)
uniquely determined, and so the multipliers
theorem are uniquely specified by
solution
n,
are obviously
and p.
Furthermore,
of the K-K-T conditions for the
show that the complementarity
of the K-K-T conditions
are met,
as well
as that
p are nonnegative.
If
i,
of optimal
BSi,
Yi,
i,
ti,
ui, ,
K-K-T multipliers,
vi, i=l,...k,
m,
n,
p are a set
then from the uniqueness of
14
(YAi
+ ZAi),
-
we must have that a i
and all quantities must
=
Si
be nonnegative,
i=l,...,k.
Similarly, we can show that
u
>
ui ,
i
vi
v i,
i=l,...,k.
and similarly ti YBi = 0, u
k
k
m =
E
so m KA = 0
n KB = 0
ai
> rA -
implies
ai
E
set of multipliers
then m = 0, and
i,
<
ai
= ai,
Corollary 2.
that
There
Note that,
specified in
and
and
the
ai = rAB
Similarly,
ti = ti,
p =
.
Thus,
i=l
O,
Si =
=
Yi,
vi = vi,
establishing the
i=l,....,k.
in particular, if K > O,
Bi, Yi
ui = ui,
K-K-T multipliers
(ai, Bi),
Also
k
in this case.
exists a unique set of optimal
satisfy Yi = max
ZBi = 0.
so m > 0 and m > m,
ai
i=l,...,k, and m = m = n = n = p =
the uniqueness of the optimal
i
and indeed are the smallest
~
so
= si,
Yi > Yi,
can take.
i=l,...,k.
and it then follows that si
ti,
> si,
YAi = 0, then
i
Thus the multipliers
i=l
since ai
i, ti
ZAi = 0, v
i
k
If K > O,
>
si
> ai,
Similarly, n > 0, p > 0,
theorem are optimal K-K-T multipliers,
values that any
i
= 0
But ai-si
whereby ai
> 0, and
that m KA = 0.
= O.
and p KAB
=
si.
because s
Thus,
Si YAi = 0,
rA -
i -
X]
K-K-T multipliers
[X]
then the optimal K-K-T
multipliers are unique and hence must satisfy Yi = max
(ai,
Bi),
i=l,...,k.
The economic significance of this corollary is clear.
ai
and Si
The
represent the marginal values of A-capacity and B-capacity,
respectively,
in state
i,
i=l,...,k.
marginal value of AB-capacity
in state
15
Furthermore,
i,
i=l,...,k.
i represents the
The condition
11
Yi
= max
(ai,
i),
i=l,...,k asserts that in each
value of AB-capacity is
the
state the marginal
larger of the marginal value of A-capacity
or of B-capacity.
In the absence of flexible capacity,
of
the firm faces the decision
how much A-capacity and B-capacity to build by solving the two
independent manufacturing investment
problems AMS
AMS:
and BMS:
BMS:
max
KA,XAi
i=1 ....
subject
k
2
E [-ai(XAi)
+ bi
i=1
rA KA +
XAi]
k
XAi -
to:
KA <
0
(i)
>
0
(si)
KA >
0
(m)
XAi
k
max - rB KB +
KB,XBi
i=l
i=
.. ,k
subject
[-Ci(XBiJ
XBi -
to:
KB <
0
(i)
>
0
(ti)
KB >
0
(n)
KBi
As is discussed in a companion paper
(Freund and Fine
+di
[1986]),
these
two problems possess unique solutions and unique K-K-T multipliers,
and can be solved analytically.
In view of corollary 2, we have the
following theorem:
Theorem 2
(Necessary and sufficient
conditions for purchasing
flexible capacity).
Let KA,
XAi,
be optimal
i,
Si,
i=1,...,k,
m,
and KB,
solutions and K-K-T multipliers
Xgi,
Bi,
ti,
=1
... ,k,
for the independent
manufacturing investment problems AMS and BMS, under assumptions
A1-A5.
Then KAB > 0
in the optimal
solution to the PFMS problem
k
and only if rAB <
E
max (ai,
i).
i=1
16
if
n
XBi]
The economic interpretation of
respective problems
values of
extra A-
represents
AMS and BMS,
production of either A or B,
The
be economical,
states,
ai
and Bi
and B-capacity in state
the marginal value
capacity.
Theorem 2 should
i.e.,
i.
Thus
the marginal
that in order
cost rAB must be less
of the marginal value
In the
represent the marginal
max(ai,
of capacity that can be used
theorem states
its
be clear.
Bi)
in
value of flexible
for flexible capacity to
than the sum, over all
of the capacity's most valuable use in
that state.
k
PROOF:
If rAB >
max
E
(i,
Si), define Yi
max (i,
Bi),
i=l
k
ui
Yi
-
ai
+
si,
Yi
vi
-
+ ti,
Bi
and p
-
rAB
Yi.
i=1
Then the K-K-T conditions
KAB
for the PFMS
problem are satisfied with
= 0.
On
the other hand,
if rAB <
k
~
max
i=l
(i,
Si),
then the above
solution is the unique solution to the K-K-T conditions satisfying
k
(1).
However,
(rAB -
Yi)
E
< 0, and so by lemma 1,
the solution
i=l
KA,
2.
KB,
XAi,
XBi,
i=l,...,k,
and KAB
Properties of the Optimal Value
We now turn our attention to
problem to
the costs of
z*(rA,
rAB)
rB,
for given
= 0 cannot be optimal.
Function
the sensitivity of the PFMS
capacity, namely rA,
be the optimal
capacity costs rA,
value
rB,
rB,
and rAB.
Let
function of the PFMS problem
and rAB.
17
[X]
Our concern with
IH
z*(rA,
rB,
rAB)
lies
value function,
in ascertaining the properties of
to predict
the savings
or costs
this optimal
due to decreased or
increased capacity costs.
Let r =
ar =
(rA,
costs.
(rA,
rB,
Then
rB,
be the vector of
rAB) denote a vector of
z*(r+Ar) = z*(rA
the optimal value
Theorem 3
rAB)
capacity costs.
changes
+ ArA, rB + ArB,
Let
in optimal capacity
rAB + ArAB) measures
of the PFMS problem when capacity costs are r +
(Characterization of
the Optimal
r.
Value Function)
Under assumptions A1-A5,
(i)
z*(rA, rB,
with
(ii)
rAB)
is a convex piecewise-quadratic function,
continuous first
If K = (KA,
KB, KAB)
capacity costs are
rA
=
partial derivatives.
are
optimal capacity values when
r = (rA,
rB,
rAB),
then
-KA
arA
arB
(iii)
where z*(r)
= -KB
2rAB
-KAB
If r =
(rA, rB,
rAB)
is
in the interior of a region
is a quadratic form, and
r
is sufficiently small, then
z*(r+Ar) = z*(r)-KA(ArA)KB(ArB)K-KAB(ArAB)
M
is a symmetric negative semi-definite matrix.
is negative definite, and so z*(r+Ar)
This theorem gives
it
- (1/2)(Ar)TM(Ar),
Furthermore,
partial derivatives given by (ii)
The interpretation of
these first partial
18
then M
is strictly convex.
us the structure of the optimal
is convex and piecewise-quadratic.
continuous first
If K > 0,
where
value function it has
in Theorem 3.
derivatives
is that profit
declines with increases in
optimal
costs
capacity value.
on op2timal
capacity costs at a rate
The effects of
small
equal to the
changes
capacity levels and consequently
in capacity
second-order. The theorem also shows
that the optimal value
is decreasing
in each capacity cost,
because KA
and KAB > O.
The formula
form for z*(r+Ar)
part of the
PROOF:
KB > 0,
and
gives us an explicit functional
(iii)
In the next section,
if we know the matrix M.
proof of Theorem 4,
O,
function
we will
show how to construct
as
the
and hence how to construct the functional form in part
matrix M,
(iii)
in
are of
on profits
the above theorem.
of
The proof
properties
of
of this theorem follows from the duality
convex quadratic programming,
see Dorn [1960].
standard quadratic programming dual of the PFMS problem PFPM2,
we denote as the DFMS problem,
can be derived as:
k
minimize
aiBii
si,t i
i=l
... ,k
E
k
(bi-ai+si)2 /(4ai)
E
+
(di-8i+ti) 2 /(4ci)
i=l
i=l
k
subject
to:
E
ai
< rA
(KA)
si
< rB
(KB)
Yi
<
(KAB)
i=1
k
E
i=l
k
E
i=1
19
rAB
The
which
III
i-si-Yi
< 0
,
i=l,
.. ,k
(ZAi)
Bi-ti-Yi < 0
,
i=l,...,k
(ZBi)
i=l,...,k
(YAi)
Si
> 0
,
ti
> 0
,i=l,...,k
ai
> 0
,
Si
> 0
Yi
V
>> 00
1
where the primal
been written in
Note that
i=l,...,k
i=l...,k
,
i=1,...,k.
variables corresponding the dual constraints have
to the right next to the appropriate dual constraint.
the capacity costs
rA,
rB, and
rAB now appear as RHS
coefficients of dual constraints whose optimal
KA, KB,
(Ygi)
K-K-T multipliers
and KAB will form a solution to the primal problem.
the theory of Lagrange
z*(rA, rB,
rAB)
(KA, KB, KAB)
duality, the optimal
is a convex function.
value function
Also,
from Lagrange duality,
are optimal K-K-T multipliers for the first three
constraints of the DFMS problem
is a subgradient of z*(rA,
rB,
if and only if
rAB),
(-KA,
see Geoffrion
the gradient of z*(rA,
rB,
rAB),
derivative, obtaining formulas
1, these
rAB.
(ii)
-KB,
[1971].
by Theorem 1, KA, KB, KAB are uniquely determined,
is
From
-KAB)
However,
so (-KA, -KB,
-KAB)
and each component is a partial
of the theorem.
first partial derivatives are
continuous
From corollary
in rA, rB, and
Finally, from the theory of parametric quadratic programming,
see Van de Panne
[1975],
quadratic function of
the optimal
value
the RHS coefficients
function is a piecewise
rA, r,
rAB of the dual
problem.
In each region where z*(r)
z*(r)
is a quadratic form, the Hessian of
exists and is a symmetric positive
20
semi-definite matrix, because
z*(r)
is convex, whereby the matrix M given in
symmetric and negative semi-definite.
follows
(iii)
of the theorem is
The formula given in
from Taylor's theorem and the fact that z*(r)
is piecewise-
quadratic and therefore has no n-order terms beyond n=2.
The proof
that M
strictly convex)
[X]
is negative definite (and hence z*(r)
when K > 0
is deferred until
(iii)
is
the next section,
and
follows as part of the proof of Theorem 4.
One
important point regarding Theorem 3 deserves further
elaboration.
The
theory of parametric quadratic
e.g. Van de Panne
[1975],
asserts that z*(rA, r,
piecewise-quadratic function.
quadratic form
(rA,
rB,
rAB)
rAB) is a convex
Furthermore, there are a finite
S 1 ,...,SJ in the space R 3 of
number J of closed polyhedral regions
values of r =
programming, see
for which the function
in each region Sj, j=l,...,J.
z*(.)
Each of
is a
these regions
can be presumed to be 3-dimensional.
The set of boundary points
B c R3
i.e.,
at which z*(.)
derivatives
measure
has no Hessian,
is given by B =
zero in R 3 .
Thus
3 is valid for all r =
J
u
aSJ,
j=1
no second-partial
and is a set of Lebesgue
the formula of assertion
(rA, r,
rAB)
(iii)
except for those
of Theorem
r
B;
and B
has measure zero.
3.
Directional
Properties of
In this section,
levels
Under
KA,
KgB,
the Optimal Capacity Function
we examine the sensitivity of the optimal capacity
and KAB to changes
assumptions A1-A5, Theorem
functions of rA, rB,
KB(rA, rB,
rAB),
rAB,
and so
and KAB(rA, rB,
in capacity costs
1 ensures that KA,
rA, r,
KB,
and KAB are
can be written as KA(rA,
rAB),
21
rAB.
and the vector K =
rB, rAB),
(KA, KB, KAB)
[II
can be written as K(rA,
"(rA,
parenthesized form
lies
rAB).
We will
rB,
in determining
function relative
to changes
rAB)"
KB,
KAB)
Our
properties of the optimal capacity
in
(rA,
aKA
arA
2KB
arA
aKAB
arA
SKA
aKB
aKA
arB
arB
)rB
arAB
aKBKA
arAB
9K
KAB
arAB
rB,
rAB).
In particular,
direction and magnitude of changes
the
derivatives,
_
give valuable
relative to changes in
(KA,
for notational convenience.
matrix of optimal capacity/cost partial
when it exists, will
to K =
refer
but will usually omit the
capacity function,
as the optimal
interest
rB,
(2)
information regarding the
in optimal
capacity levels
capacity costs.
Corollary 3 (Characterization of the Optimal Capacity Function).
Under
assumptions A1-A5,
if r =
is a quadratic
a region where z*(r)
optimal
capacity values for
K(r+r
= K(r) + M(Ar),
is the matrix given in
for
(iii)
in
PROOF:
This result
gradient
Hessian
of
is
rB,
rAB),
in the interior of
(KA,
KB,
KAB)
are
then
r sufficiently small, where M
of Theorem 3.
In particular, M
is
capacity/cost partial derivatives
(2).
defined
(ii)
rAB)
form, and K =
r = (rA,
precisely the matrix of optimal
parts
(rA, rB,
and
is an immediate
(iii).
z*(r),
of z*(r).
From part
consequence of Theorem 3,
(ii),
and-the matrix M is
Thus,
for example,
22
(-KA,
-KB,
-KAB)
is
the
precisely the negative of the
MAAB =
MA,AB
z Az*/r
r
BrAB arAB
a rA
A
_
KA
arAB
and similarly,
9KA
MA, A
=
MA
=
B
,
+ M(tr)
for
=
is piecewise
linear,
compute the matrix M.
the
(2).
whereby K(r+Ar)
=
[XI
we will show how to
proof of the next theorem,
As part of the
M is
derivates given in
sufficiently small.
r
Therefore,
AB
capacity/cost partial
From Corollary 1, K(r)
K(r)
MAB,AB
9,
=
and MB,AB
'rA·
9rr B
matrix of optimal
MB B
corollary 3 can be viewed as
Hence,
constructive.
section is:
Our major result of this
Theorem 4
(Optimal Capacity/cost Directions and Magnitudes).
assumptions A1-A5,
(1)
if K(r)
>
0,
strictly
then
decreasing
rA
KA
is
(B)
KB
is strictly decreasing in rB
(AB) KAB is strictly decreasing
(2)
in
(A)
in rAB
(AB) KA is
strictly increasing in rAB
KB is
strictly increasing in rAB
(A)
KAB
is strictly
increasing in rA
(B)
KAB is strictly
increasing in rB
23
Under
(3)
(4)
(5)
(6)
(A)
KB
is strictly decreasing
in rA
(B)
KA is strictly decreasing
in rB
(A)
KA + KAB is decreasing in rAB
(B)
KB + KAB is decreasing
in rAB
(A)
KA + KAB is decreasing
in rA
(B)
KAB + KB is
(A)
KB + KAB
(B)
KAB
increasing in rA
is decreasing
+ KA is
in rB
increasing in rB
Before proceeding to the proof of this theorem,
interpretation of
this theorem and its
Taken together,
rAB
statements
we present an
immediate consequences.
()-(AB)
and
(2)-(AB) assert that as
increases, the firm will purchase less flexible capacity and
more of
each type of nonflexible capacity.
(2)-(A), and
purchase
Statements
(3)-(A) assert that as rA increases,
less A-capacity, more AB-capacity and
Statements
(1)-(A) and
the firm will
less B-capacity.
(2)-(A) are quite intuitive.
follows because the substitution of flexible
(1)-(A),
Statement
(3)-(A)
capacity for A-capacity
induced by an increase in rA also reduces the value, and hence the
need,
for B-capacity,
B-capacity.
the above
because flexible capacity also substitutes for
Statements
(1)-(B),
statements, for product
Assertions
(4),
(5)
and
(6)
(2)-(B) and
B.
of
the theorem indicate a
effect due to changes in capacity costs.
as the cost of AB-capacity is
(3)-(B) are analogous to
increased,
24
"ripple"
According to statement
the decrease in flexible
(4),
capacity is larger in magnitude than the increase in either type of
nonflexible capacity.
has a similar
(5)
Statement
interpretation.
the cost of nonflexible A-capacity is
(a decrease)
change
increased,
when
For example,
the magnitude of the
in A-capacity is the
largest, followed by the
(an increase), and
then by the change in B-
change in AB-capacity
capacity (a decrease).
The A-capacity is
the B-
the most affected,
capacity is the least affected, with the AB-capacity falling between
the two
nonflexible capacities.
Statement
The "ripple"
to statement
of the theorem is analogous
(6)
effect is A
to AB
(5),
to B.
for
product B.
Before presenting the formal proof of Theorem 4 we first
preceded by an explanation of the underlying
three antecedent lemmas,
economic and mathematical
K(r+ar)
from corollary 3,
concepts used in
= K(r) + M(Ar)
B, where B
E
finitely many
polyhedral convex sets sj
Recall that
r sufficiently small
the boundaries of the
in the space of
rER 3 ).
is continuous in r under assumptions A-A5, it suffices
to prove Theorem 4 when
becomes a proof of
rB, and hence
becomes MA,A < 0,
to prove
the proof
of Theorem 4
certain properties of the matrix M of optimal
capacity/cost partial derivatives.
order
the proofs.
for
is the union of
(except for r
Because K(r)
present
statement
For example, statement (1)-(A)
(4)-(B) becomes MB,A + MAB,A < 0,
etc.
In
the theorem, then, we will explictly construct the
matrix M.
As a means toward computing the matrix M of optimal capacity/cost
partial derivatives,
subproblems P(i),
solution to P(i)
we will proceed by first solving the state
i=l,...,k, and then examining the parametric
as K changes.
We
25
then will use
this analysis to
11
construct a matrix N that will
AK
= K(r+Ar) ,-
K(r) for all
satisfy -N(AK)
sufficiently small
Theorem 3 and corollary 3 is given by M =
the
sign and magnitude
Thus
P(i),
r,
r.
-N- 1 .
where
The matrix M of
We
then will
Let K =
values for capacity costs
Note that the
(KA, KB,
KAB)
r =
rB,
(rA,
subproblems
capacity
be the optimal
rAB)
satisfying assumptions
economics of subproblem P(i)
implies
marginal profit from producing an extra unit of product A
decreasing in XAi
=
(YAi + ZAi) and
zero when at XBi = di/2ci.
subproblem P(i)
Figure 2 plots
the optimal
six regions
bounded by the inequalities
regions numbered #1-#6,
in
The optimal
the space of
the optimal
is given according to Table
and di/2c i
KB,
(bi/2ai) and
in the figure.
(di/2ci),
In each of
natural
Region
economic
1.
1, which also gives
the optimal
As
figure
in Table
1
interpretation as follows:
is enough capacity KA, KgB,
and KAB
to attain bi/2ai and B production to attain di/2c
Therefore, the capacity constraints in
binding, and
K-K-T
Each of the six regions of Table 1 has a
In this region there
for A production
the
solution to the subproblem P(i)
the regions #1-#6 given by the inequalities
are nonoverlapping.
and
and KAB-
multipliers and the algebraic description of each region.
2 indicates,
i.
solution to the
capacity values KA,
indicated
is
profit is decreasing and
will depend on the quantities bi/2ai
their relationship to
that the
= bi/2a
reaches zero at XAi
Similarly, for product B, the marginal
reaches
examine
of specific elements of M.
our first task will be to solve the state
i=l,...,k.
A1-A5.
=
subproblem P(i)
are not
there is no need to allocate scarce capacity.
26
i.
Region 2.
marginal
In this region,
profit
there
(ai = 0),
from product A
capacity for product B to
is ample A-capacity to attain zero
but not enough B-
attain zero marginal
profit.
and AB-
Thus product
B uses up all of the B- and AB-capacity, which will have positive
shadow values at the
Region
3.
optimum
In this region,
(i.e.,
There
marginal value
is devoted to
There is
but not enough A- and AB(i
=
Yi > 0).
and AB-capacity.
or B-capacity to attain zero
Furthermore,
A-production
insufficient A-
the flexible AB-capacity.
so dominates
the marginal
that of B-production
marginal value of either product's
B-production
0),
B-capacity to attain zero
of either product's production without resorting to
A-production so dominates
Region 5.
the A-
is insufficient A-
the flexible AB-capacity.
capacity
(Bi =
> 0).
A to attain zero marginal value
Thus product A uses up all of
Region 4.
Yi
there is ample
marginal profit of B production
capacity for product
=
Bi
(ai =
Yi >
profit of
that all
i >
flexible
0).
or B-capacity to attain zero
production without resorting to
Furthermore,
the marginal value of
that of A-production
capacity is devoted
to B-production
(Bi
Region 6.
insufficient A-
or B-capacity to attain zero
There is
marginal value
the
of either product's
flexible AB-capacity.
Both
=
Bi
=
Yi
>
i
> ai
> 0).
production without resorting to
products share the AB-capacity and
the marginal value of A-production
(lti
=
that all flexible
0).
27
and B-production are the same
(di/2ci )
Region #5
Region #2
I
cm
-
Region #6
di/2ci = KB
KB
+
KAB
KAB
'b.>0
x
Region #4
= KB
KB
Region #1
om
Region #3
II
-4
ci
-4
.
KA - KAB
KA
Figure 2 - The Six Regions of Lemma 2
2I
(bi/2ai)
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'P'4
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O
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N
+
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NM
IO
+
O
+
N
_
*_
for subproblem P(i).
optimal
the
Furthermore,
the K-K-T multipliers s i ,
With
r =
(rA,
rB,
ti,
u i,
rAB) given,
(ArA, ArB,
rAB) be a vector of
Let
K = K(r+Ar)
-
the
Knowing
1.
unit change
unit changes
per unit
,KAB-
allows us
the data for state
= bi-2aiKA -
in
i,
Bi,
changes
1) are
in ai,
For example,
i results
in
Bi,
in the bottom row of Table
N
Yi
from
di )
2c i
(
2a
I
2aiKAB, and the per
These per
KAB
let the matrix Ni denote the negative
and vi relative
to a change in KA, KB,
is the negative of
i relative to a change in KAB.
i) of the negative
,
shown in Table 2.
the element n,AB of Ni
By definition, nB,AB,
to easily compute
Vi relative to changes in KA, KB,
each subproblem P(i),
unit change
capacity/cost partial
in ai relative to a change in KAB is -2ai.
(derived from Table
For
(AKA, AKB, AKAB)
then ai
Let
capacity values
the subproblem shadow prices ai,
if
For example,
falling in region #4,
=
K
in
satisfy
According to corollary 3,
r.
is the matrix of optimal
where M
i
Bi and
be the change in the optimal
K(r)
per unit changes
Table
ai,
small capacity cost changes.
induced by the change in capacity costs
derivatives.
vi,
in the
each region
let us assume that rB.
Ar =
AK = M(ar),
in
1.
lemma
(1) of
formulas
1, and so the solutions given are
in Table
satisfied for each region
table,
verify that the K-K-T conditions are
is straightforward to
It
2.
=
the per
This denotation appears
Define the matrix N by
k
E
Ni
i=l
for example,
.
represents
per unit change in Bi
31
(3)
the sum
(over all states
relative to a change in KAB,
III
.)-
o
0
N
C.)
04
*
C.)
c1
cu
+
0
+
C.)
o
c
1
4
-
)
0
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cu
II-
-c
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.1 cl
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C.)
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.
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Wr
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to.o+
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(d
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c.j
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0bO
4*
Ci
4*
In
4*
4*
CO
4*
*
a)
a.)
04)
CM
32
k
i.e.,
nBAB is
the negative
£
change in the
Bi
relative to a
i=l
k
If KB >
change in KAB.
0,
then
£
Bi =
rB,
and so nB,AB is
the
i=1
negative
per unit change in rB relative to a change in KAB.
similar logic for the other components of N,
Lemma 2.
K(r) >
Under assumptions A1-A5,
if r =
Using
we have:
(rA,
rB, rAB)
B, and
0, then the matrix N defined by Table 1, Table 2,
and
(3),
satisfies
N(AK)
If the
matrix N
Lemma
3.
Table
2,
is invertible,
-N - 1 .
must have M =
= -(ar)
Thus,
[xl
.
= (-N-
then aK
our next task is
)(Ar), whereby we
to prove
Under assumptions A-A5, the matrix N defined
and
(3) has a positive determinant.
matrix N can be written as
N
1
=
e+b
e
e+a
e
e+d
e+c
e+a
e+c
e+a+c
,
where
E
a =
iER
2ai
UR
3
4
E
2a
iER 3 UR4 UR 5
b
=
c
=
£
R 2 uR
d =
iER
e=
i
2Ci
5
E
2ci
uR
uR
2
4
5
2aici
E
iER
6
ai+c
33
(4)
by Table
Furthermore, the
1,
III
and R
in region j
as defined
in
j=1,...,6.
Table 1 (or Figure 2),
PROOF:
i that are
the set of state
denotes
See Appendix.
[XI
By Cramer's rule, we obtain:
Lemma 4.
K(r)
Under assumptions A1-A5,
> O,
det(N)
r = (rA,
if
rB,
rAB)
B, and
then
nB,AB nB,AB -nB,B nAB,AB
nA,B nAB,AB -nA,AB nB,AB
nB,B nA,AB -nA,B nB,AB
nA,B nAB,AB -nA,AB nB,AB
nA,AB nA,AB -nA,A nAB,AB
nA,A nB,AB -nA,B nA,AB
nA,A nB,AB
nA,B nA,B
nB,B nA,AB
-nA,B nB,AB
-nA,B nA,AB
-nA,A nB,B
[X]
With
the formula for M given
in lemma 3,
Theorem 4, which describes how optimal
we are
in position to prove
capacity levels change as a
function of the capacity costs.
Proof of Theorem 4:
and from equation
Given the formula for the matrix M from lemma 4
(4),
the proof of Theorem 4 is accomplished by
verifying certain inequality relationships among
matrix M.
The details of this procedure are
in the Appendix
Theorem 4 is conditioned on the supposition
capacity values are positive.
values,
is
say KA,
is
that all
the
[X].
optimal
When one of the optimal capacity
zero, we obviously can no
strictly decreasing
the elements of
longer assert that KA
Furthermore, the computation of
in rA.
matrix M cannot be accomplished as
the
in the proof of Theorem 4,
k
because we may not have
£
a
i
= rA,
i.e.,
m could be positive.
i=l
However, using logic similar to that in Theorem 4,
Corollary 4:
4
Under assumptions A1-A5,
the six assertions of
remain valid, without the strict monotonicity
34
we can prove:
in assertions
Theorem
(1)-(3).
4.
Example with Calculation of the Value of Flexibility
Consider a firm that produces two basic
and faces uncertainty about
If this
firm must make its
product lines,
A and
B,
the future demand for each product line.
investment decisions on flexible and on
nonflexible capacity levels before the demand uncertainty is resolved,
then our model provides a useful tool
decision
problem.
industry.
for analysis of the investment
One instance of such a setting occurs in the auto
In their engine plants, automobile manufacturers
choice of building nonflexible machining
one size engine block or
between four and
lines that can only produce
flexible lines that can switch,
six cylinder engines.
cars with small or
movements
large
for example,
Since engine plant
construction lead times and useful lives are
frequency of oil price
face the
long relative to the
(which affect consumer demand for
engines),
automobile manufacturers may have
an incentive to build
some flexible engine line capacity that can be
used for either large
or small engines.
The following example
automobile
illustrates the calculations that such an
manufacturer could perform to analyze
manufacturing system investment decision.
for this example,
its
We will
product-flexible
also illustrate,
a calculation of the value of flexibility as a
function of the per unit cost of flexible capacity.
The value of
flexibility as a function of rAB is measured by z*(rA,
z*(rA,
rB,
is available
do not
),
the additional
profits obtained when flexible capacity
at a cost of rAB per unit.
reflect actual
-
rB, rAB)
The numerical data presented
industry conditions.
They are only used
for
purposes of illustration.
Suppose there are four possible future states
depending on whether oil
of the world,
prices are high or low, and whether
35
the U.S.
III
growth is high or low.
gross national product
for products A and B
(small and
suppose demand
linear
large engines) is
respectively, and that the state of
of A and B,
the notation of
the previous
and di=pig i,
product
demand slopes are unity, i.e.,
The data
Oil
Prices
for i=1,2,3,4.
For
Pi
in
simplicity, we assume the
fi=hi=l
for the example are as
GNP
Growth
Thus,
sections, the random variables are
bi=Piei
ai=ci=Pi).
in the prices
the world only affects
intercept of the products' demand functions.
the vertical
St ate
i
Further,
Small engine
(Product A)
demand
intercept, e i
for all i (so that
follows:
Large engine
(Product B)
demand
intercept, gi
Large a nd
Small E ngine
demand slope
fi, h i
1
High
High
.10
18
10
1
2
High
Low
.30
16
6
1
3
Low
High
.40
8
18
1
4
Low
Low
.20
4
16
1
Further, we assume the per unit capacity costs are
rB
= 2.2,
The
rAB
= 3.8.
table below shows
terms of the quantities
State
i
rA = 2.6,
ai
ai,
i
the data
bi,
for each state expressed
ci,
in
di.
ci
di
1
.10
1.8
.10
1.0
2
.30
4.8
.30
1.8
3
.40
3.2
.40
7.2
4
.20
.8
.20
3.2
Clearly, the assumptions Al A5 are satisfied,
problem exhibits a unique solution.
36
and so this PFMS
This optimal solution is
KA =
3, KB = 4,
KAB =
3.
the two
However,
as the
to hold
less
some
the uncertainty in the demands for
products make purchase of 3 units
flexible capacity efficient.
well
non flexibile capacity costs
the firm optimally chooses
than flexible capacity,
nonflexible capacity.
Since
The dual
region (from Figure 2
of the more
prices a i ,
in the previous
Bi,
expensive
and Yi,
section)
as
for each
state are given below.
ai
State
i
i
Region #
1
.6
.2
.6
4
2
1.2
0
1.2
3
3
.8
1.6
1.6
5
4
0
.4
.4
2
Straightforward calculation shows
that z*(2.6, 2.2,
order
to compute the value of flexibility,
+c)
50 1/12 by solving the PFMS problem will
=
the value of flexibility
In order
in
(when rAB = 3.8)
to obtain N,
= 52.5.
we compute z*(2.6,
rAB = +.
is 52.5 -
the matrix of negative
2 to first calculate Ni
for each state
procedure, we obtain
37
·11__^1______11____ .._
In
2.2,
Therefore,
50 1/12 = 2 5/12.
per unit changes
the capacity shadow prices relative to the capacity levels,
use Table
_11_
3.8)
i.
we
Following this
II
.2
0
.2
.2
and M
=
=
-N
0i
- 1
=
=
.4
.4
.4
,
-17/16
3/4
-3/4
7/8
-2
1 .5
1.5
7/8
.6
J
0
0
0
.8
.8
LO
.8
.8
=
1.6
0
.8
0
1.4
1.2
_ .8
1.2
2
N =
and
N3
,
o
.6
0[
.4
.8
.6
0o
O
.2
0
N4
, N2
O
0
o
.6
.2
-1.75
Thus,
z*(2.6+ArA,
2
.2 +ArB,
+ 1/2
rA
ar
3
. 8 +ArAB) =
T
-17/16
3/4
B
\&rAB
In particular,
3
52.5 -
7/8
(ArA)
-3/4
7/8
-2
1.5
1
-1.75
1.
v
-
4(ArB)
/
-
(ArAB)
rA
ar
B
ACrAB
rB /
for example
az*
arAB
=
-3
and
That is, with an incremental increase
aKA
A
arAB
77
8
in the cost of flexible capacity,
optimal profits decrease at the rate of three units, and
dedicated A-capacity increases at the rate of
5.
3
optimal
.875 units.
Discussion
Our model
provides the beginnings of a conceptual framework for
evaluating investments in
To ease
product-flexible manufacturing systems.
the analysis of the model, we have assumed that the firm
produces only two different products and
38
can invest in only three
different technologies.
An alternate formulation, due to Kulatilaka
[1985], allows M different types of nonflexible technologies,
of which may be optimal
each
in certain states; and analyzes the optimal
use over an infinite horizon of a flexible technology whose "mode"
can be switched each period (at a cost) so as to emulate any of the
M nonflexible technologies.
The modelling cost of this richer
formulation is that Kulatilaka makes a very specific assumption on
the distribution of the sequence of random variables relevant to the
optimal single-period technology choice.
(The random process is a
discrete version of mean-reverting Weiner process.)
Kulatilaka is
able to calculate optimal policies for specific numerical examples.
In contrast, our model, which we restrict to only two products,
assumes that the firm may invest in a portfolio of flexible and
nonflexible technologies and that the switching costs are either
zero (for the flexible technology) or infinite (for the nonflexible
technologies.)
Our model formulation PFMI can easily accommodate
multiple time periods by adding a time index to XA i, XBi,
fi,
gi,
hi,
Pi,
ei,
summing the profits over the time periods in the
objective function, and adding a discount factor.
The analysis
remains the same, assuming the firm does not carry inventories from
period to period.
If we permit inventories, then the analysis of
the model becomes significantly more difficult.
Our intuition is
that the presence of inventories would reduce the value of flexible
capacity to the firm since holding inventories provides flexibility
for firms in meeting demand.
We think that an extension of the
model, exploring this conjecture, warrants further analysis.
Another extension of our model would be to assume that the per
unit variable cost of production differs, depending on whether the
39
1
_1___1__1___11·1__1__--_I^_____
_--
HI1
unit
is produced with the flexible or nonflexible
Many
capacity.
analysts claim that automated flexible manufacturing systems
than do their less flexible
lower per unit variable costs
When this
substitutes.
have
is the case, the firm will obviously buy
more flexible and less nonflexible
for the formulation analyzed here.
capacity than would be the case
Suppose,
in addition, we
generalize the form of the acquisition cost to include
a fixed
component if any positive level of a certain technology is chosen.
assume that investing in KAB > 0 units of flexible
For example,
capacity costs RAB + rAB.KAB, where RAB > 0.
nonflexible
capacity investment costs
we
are RA + rAKA and RB + rBKB.
increase the difference between
the variable cost of producing
with the flexible versus the nonflexible
increase RA,
RB,
KA and KB will
purchased.
and KAB as
levels of KA, KB,
if we look at the optimal
In this case,
Similarly, assume
technology or as we
and RAB, then at some point
the optimal
levels
become zero and only flexible capacity will
Thus,
a more realistic
of
be
production cost and acquisition
cost structure can yield a solution where the option of investing in
nonflexible capacity
is abandoned completely in favor of
investment
We think that an exploration of this
solely.in flexible capacity.
extension merits further work.
A
final
direction for further exploration of this
look at how competition affects
flexible capacity.
firms'
optimal
investment
is viewed as threatening to enter market A
in
[1979],
Bulow et al.
40
If firm 2
(only) then firm 1 may
want to hold some excess A-capacity to deter entry.
Dixit
to
In one two-firm extension of our model, suppose
that firm 1 is a monopolist in product markets A and B.
Spence [1977],
model is
[1985].)
(See, e.g.,
The question of
interest is whether firm 1 can deter entry by holding productflexible
(KAB) capacity.
Holding KAB to deter entry is
less costly
than holding only KA to deter entry because unused KAB capacity has
an alternate use.
However,
KAB may not be a credible
The
because of
this alternate use,
holding
threat to potential entrants.
crucial issue here relates
commitment and flexibility.
to the conflicting effects
of
To deter entry, an established firm
must appear
to the entrant to be committed to behavior that will
detrimental
to a new entrant.
credible,
for example,
that marginal
in a new market
[1983]).
Such a commitment can be made
by having already
costs of production are
Alternatively,
be
invested in capacity so
low (as
in Dixit
1980]).
a firm may demonstrate commitment by investing early
(as in Spence
[1979],
and Fudenberg and Tirole
Holding product-flexible capacity would appear to reduce
the credibility of commitment to high post-entry production.
is because,
once entry has
occurred,
have a higher marginal product
multimarket monopolist who
is
This
the flexible capacity might
if deployed elsewhere.
Therefore a
threatened by potential entrants may
not wish to hold flexible capacity because
it detracts from the
credibility of commitment to defend the threatened market.
Countering this effect,
prefer to hold some
the previous
however,
a multimarket monopolist will
flexible capacity if demand
sections)
uncertain because the
or if entry is uncertain.
potential entrant's
[1983],
Milgrom and Roberts
Further analysis
(Entry may be
Kreps and Wilson
[1982],
is required to
resolve how flexible technology affects equilibrium capacity
investment when these
two conflicting forces
41
(as in
costs are not observable
by the incumbent as in Saloner
[1982a].)
is uncertain
are present.
and
III
Similar questions revolve around how product-flexible capacity
might be used by potential entrants.
In particular, purchasing
product-flexible capacity for entering a new market may be a less
risky way to enter a market controlled by an
costs.
That is,
Eaton and Lipsey
incumbent with unknown
low exit costs may reduce the costs of
[1980]
for additional
entry.
perspective on this
(See
issue.)
However, an incumbent might play tougher against an entrant who is
known to have product-flexible capacity that can be used in other
markets.
(This
is especially likely if
opportunity for reputation formation as
or Milgrom and Roberts
[1982b]).
the incumbent has an
in Kreps and Wilson
Obviously, there
[1982]
is much work to
be done to determine how the existence of product-flexible capacity
affects competition and entry in this multi-market oligopoly entry
game.
42
Appendix
PROOF
of Theorem 1:
The PFMS problem is always feasible, since
the PFMS problem is a convex quadratic
A2 and A3 it is bounded from above.
Because the objective function is
XAi and XBi,
i=l,...,k, XAi,
It remains
to
by assumptions
strictly convex
in
XBi are uniquely determined
Assuming the contrary, suppose
2
2
2
(KA, KB, KAB).
program, and
It thus attains its optimum.
show that KA, KgB,
alternative optimal
By assumption A2,
is feasible.
i=l,...,k, KA=KB=KAB=O
XAi=XBi=,
i.
KAB are uniquely determined.
1
1
KA, KB,
values of KA,
for each
1
2
2
KAB and KA, KB,
KB, and KAB,
and
2
KAB are
1
1
1
(KA, KB, KAB) *
1
1
1
2
2
2
Then rAKA + rBKB + rABKAB = rAKA + rBKB + rABKAB.
We have two cases:
Case
1.
1
2
KAB = KAB.
1
2
that KA > KA > 0,
i=l,...,k, m,
Then XAi
2
< KB, and
n,
1
Then KA
2
1
and hence KB > KB
p)
be any optimal
KA < KA, whereby
so ei =
2
1
KA and KB *
,
>
0.
2
K,
Let
= O.
i = 0, i=l,...,k.
rA =
rAB
Xi
and rB =
1
KA
then rAB =
2
then KAB = KAB = 0.
2
KA and XBi
X,
and
i=1
+ P-
If p = 0,
But
2
and KB > 0,
k
Xi,
i=1
k
=
E
i=l
ri,
Similarly XBi < KB
1
Also, since KA > 0,
=1,...,k.
Thus
ei, Xi, qi,
(i,
K-K-T multipliers.
k
m = O and n
and we can assume
KB
rA, violating assumption A4.
For each
1
+ KAB = KB,
+ KB + KAB, and so Xi =
index i=l,...,k,
and so XAi
+ XBi
, i=l,...,k.
contradiction.
43
XAi
Thus p > 0.
2
2
< KA + KAB
2
1
1
1
KA + KB < KA + KB =
Thus rA = O,
a
=
I[
Case 2:
KAB
1
2
KAB > KAB
> 0,
addition.
(ei
Without
and so
i
Then
-.
k
£
KAB.
+ Xi)
=
that
1
1
2
2
Suppose that KA + KAB
KA + KAB, in
k
i=l,...,k, whereby rAB =
E
(Si + ei + Xi)
j~~~~i=1
0,
n.
i=l
assumption A4.
of generality, we can assume
p = 0.
~~~~~~
= rB -
loss
This
1
implies that rAB
1
2
rB,
=
violating
2
Therefore KA + KAB = KA + KAB, and by an identical
argument,
it must be true that KB + KAB = K + KAB.
Therefore,
2
1
1
2
1
2
However, it is also true that
KB = KA - KA.
KB
KAB - KAB
1
2
rAB
(KAB
KAB)
-
rA (KA
=
KA)
+ rB
(KB -
Kg).
This forces
rAB = rA + rB, which violates assumption A5, proving the theorem.
PROOF of
Lemma 3:
We note the
following inequalities as a consequence
of Table 2:
i
i
nAA
nA,AB
i
i
nBB > nB,AB
i
nAB,AB
i
nAB,AB
i
> nA,B > 0
i
nB,A > 0
i
nA,AB
> 0
i
> nB,AB > 0
i
i
nA,B = nB,A
i
i
nAB,A = nA,AB
i
nAB,B =
- which
i
nB,AB
imply:
nAA
> nA,AB > nA,B
nB,B
> nB,AB
> nB,A
0
0
nAB,AB > nA,AB > 0
nAB,AB > nB,AB
nA,B
-
nB,A
nAB,A
=
nA,AB
nAB,B
=
nB,AB
[X]
> 0O
O
44
Note,
in particular,
symmetric and nonnegative.
set of states
denote the
j=i,...,6,
in Table
that N is
i that are
in
Now let
Rj,
region j as defined
1 (or Figure 2).
Define the following numbers:
a =
2ai
E
iER 3 UR
b =
iER
c
4
E
2ai
UR
UR
5
3
4
£
=
2ci
R 2 uR 5
d =
iER
e =
Then the
_
2ci
UR
UR
2
4
5
E
iER
2aici
2asc
ai+ci
6
matrix N can be written
N =
e+b
e
e+a
e
e+d
e+c
e+a
e+c
e+a+c
in
the form:
(4)
The determinant of N is:
det(N)
= ad(b-a)
Because a,
b,
of the above
+ bc(d-c)
c, d,
terms
+ e
e are all
(b-a)(d-c)
nonnegative,
are nonnegative,
to show that some term is
=
+ c(d-c)
and b > a,
whereby det(N) >
+ ac].
and d > c,
O.
It remains
Therefore
derive a contradiction.
0 and bc(d-c)
= O.
Then one of the following
nine cases must be true:
Case 1.
or 6.
a =
But
O,
then,
b =
O.
Thus all states
by Table
all
strictly positive.
Assuming the contrary, we will
we assume that ad(b-a)
+ a(b-a)
1,
i
=
must be in
Yi for all
contradicting assumption A4.
45
regions
1, 2,
i, whereby r B =
rAB,
M11
Case 2.
a = 0,
By Table
1,
c = 0.
i =
Thus all
states must be in
Yi for all i, whereby rB = rAB,
regions
1 or 6.
contradicting
assumption A4.
Case
a = 0, d = c.
3.
5, or 6.
By Table
Thus all states must be in
1, B
=i Yi
for all
i, whereby r B
regions
1, 2,
= rAB,
contradicting assumption A4.
Case 4.
d = 0,
By Table
1, ai
b = 0.
= Yi,
Thus all states must be in regions 1 or 6.
for all i, whereby rA = rAB, contradicting
assumption A4.
Case 5.
or 6.
d = 0, c = 0.
By Table
1,
Thus all states must be in regions 1, 3,
i = Yi,
for all
i, whereby rA = rAB,
contradicting assumption A4.
Case 6.
d = 0,
d
Case 7.
b = a,
b = 0.
This is
Case 8.
b = a,
c = 0.
Thus all states must be regions 1, 3,
1, ai =
Yi,
or 6.
By Table
=
c.
This
is
identical
to case 5.
identical to case 1.
for all
4,
i, whereby rA = rAB,
contradicting assumption A4.
Case 9.
b = a, d = c.
3,
or 6.
(i
+ Bi = Yi,
If e = 0,
Thus all states must be in regions 1, 2,
then all states are
for all
i.
This
contradicting assumption A5.
a = 0 or c = 0.
regions
1, 2,
If a = 0,
5, or 6, and
this
implies all
rA =
rAB, a contradiction.
Thus det(N)
implies
Thus,
in regions
e > 0, and det(N)
so rB = rAB,
regions
> O.
or 3,
and
that rA + rB = rAB,
then this implies
states are in
1, 2,
= eac.
all states are
a contradiction.
1, 3,
4,
or 6,
If
Thus
in
c = 0,
and so
[X]
46
Proof
of Theorem 4:
det(N)
=
1.
(1)
Without
We prove each
loss of generality,
of the six assertions
We must show that
AA < 0,
Because nBB
nB,AB and nAB,AB
nAB,AB) < 0.
If
or (iii)
A,A = 0,
nB,B = nB,AB = nAB,AB1 or 3,
regions
1, implying rA = rAB = 0.
regions
results
0,
1, 2,
5, or 6,
If
If
We must show that
0.
m
(ii),
(iii),
B,B < 0
A,AB >
,
B,AB
nB,AB)
>
0O
then all
nAB,AB = 0,
in
Since
in
states must be
each case
mA,A < 0.
Similar
AB,AB < 0.
> O,
AB,A
nA,B,
> 0,
A,AB =
If mAAB = 0, then either ng,gB
= nA,B-
nB,B
then all states are
and
Because nB,B > nB,AB and nA,AB
- nA,B
(ii)
then all states are
implying rB = rAB.
or nB,B = nB,AB and n A ,AB
rAB,
(i),
(nB,AB nB,AB -
nB,B = 0,
in a contradiction of assumption A4,
(2)
nA,AB
If
(i)
separately.
and mAB,AB < 0.
nB,AB, mA,A =
implying rA = rAB.
logic can be used to deduce that
mAB,B >
mBB < 0,
then either
regions
in
we can assume
The first
and
(nB,B
= 0,
n A ,AB
case implies that
the second implies thar rB = rAB, and the third
=
rA
implies that rB-=
rAB.
Thus we obtain a contradiction to assumption A4,
Similar logic
M is
symmetric,
(3)
nAB,AB a >
be
A4.
can be used to show that mB,AB
m
nA,AB nB,AB)
> 0,
in regions
1,
Similarly,
contradiction.
> 0.
0.
Furthermore,
since
A,AB = mAB,A and mB,AB = mAB,B-
We must show that
0 and c
and mA,AB >
by
=
(e(e+a+c)
(4).
0,
-
if a = 0,
whereby rB = rAB,
otherwise
Thus -ac < 0,
B,A < 0.
(e+a)(e+c))
Furthermore,
2, 5 or 6,
c >
A,B < 0 and
and
rA =
< 0.
=
(-ac) < 0,
(nA,B
because
then all states must
contradicting assumption
rAB, which again
so mA,B
47
=
mA,B
is a
II
(4)
Note
We must
that +
AB,AB + mA,AB =
nA,B nB,AB).
But -nA,A
nA,A + nA,AB)
nB,B(-
O because b > a.
shows
(5)
nB,B
We must
(nAB,AB
c(d-c)
> 0,
(nB,B
-
-
equal
nB,AB nB,AB
+ nB,AB
+ nB,B
-
nA,AB -
nA,B nB,AB
-
= -(e+d)(b-a)
nA,B)
e(c)
=
< 0
A parallel argument
mA,AB > -m B,A
+
(nA,B
Thus
nA,B nB,AB
-
.
-
-
> 0,
48
+ (e+c)(-c)
=
But this
all components of which
i.e.,
identical to that of
A
nB,B nA,AB =
nA,AB nB,AB).
+ ea + a(d-c),
-mA,
Second, mA,AB + mA,B
-mA,A > mA,AB.
+ mA,B
First,
nB,AB) = (e+d)(c)
+ nA,B nAB,AB
to e(d-c)
is
A,B
B
0.
< 0.
Thus m A ,AB
This proof
and mB,AB + mAB,AB
+ nB,B nA,AB
mAB,AB + mA,AB > 0.
nA,B nB,AB
are nonnegative.
(6)
nA,B(nB,AB
because d > c.
second term is
+ nA,B nA,B
< 0
+ nA,
A,A nB,B
show that -mA,A
nA,AB)
nA,AB -
(-
nB,B
+ mAB,AB
= nB,B nAB,AB -
mA,AB
-
Thus
that mB,AB
+ mAB,AB
show that mA,AB
mA,AB >
(5).
[X]
mA,B-
REFERENCES
A Selective Review of the
Managing Flexibility:
1985.
Adler, P.
Challenges of Managing the New Production Technologies'
Mimeo, Department of Industrial
Potential for Flexibility.
Engineering and Engineering Management, Stanford University.
Classification of Flexible Manufacturing Systems.
1984.
Browne, J.
The FMS Magazine, April.
"Holding Idle
1985.
Bulow, J., J. Geanokopolos, and P. Klemperer.
Capacity to Deter Entry," forthcoming, Economic Journal, March.
The Fundamental Principles of Flexibility in
1982.
Buzacott, J.A.
Proceedings of the First International
Manufacturing Systems.
Brighton, U.K.
Conference on Flexible Manufacturing_Systems.
"A Model of Duopoly Suggesting a Theory of Entry
1979.
Dixit, A.
Barriers," Bell Journal of Economics, Spring, pp. 20-32.
Dorn,
W.S.
Math.
1960.
18, pp.
"Duality in Quadratic
155-162.
Programming," 9uart._ARR1.
1980.
"Exit Barriers as Entry Barriers,"
Eaton, B. and R. Lipsey.
Bell Journal of Economics, Autumn, pp. 721-729.
"An Algorithm for Computing
1986.
Freund, R.M., and C.H. Fine.
Optimal Capacity Levels for the Product-Flexible Manufacturing
System Investment Problem," in preparation.
"Capital as Commitment:
1983.
Fudenberg, D. and J. Tirole.
Journal of Economic
Deter
Entry,"
Strategic Investment to
Theory, 31, pp. 227-250.
A
"Duality in Nonlinear Programming:
1971.
Geoffrion, A.M.
Simplified Application-Oriented Development," SIAM Review 13
pp. 1-37.
1984.
Jones, R.A., and J.M. Ostroy.
Review of Economic Studies, LI,
Flexibility and Uncertainty.
13-32.
"Reputation and Imperfect
1982.
Kreps, D.M., and R.B. Wilson.
Information," Journal of Economic Theory, Vol. 27, pp. 253-279.
1985.
Kulatilaka, N.
Unversity School
The Value of Flexibility.
of Management.
Mimeo, Boston
An
Flexibility in Decision Making:
1978.
Mandelbaum, M.
Exploration and Unification, Ph.D. dissertation, Department of
Industrial Engineering, University of Toronto, Ontario, Canada.
"Limit Pricing and Entry
1982a.
Milgrom, P.R. and D.J. Roberts.
An Equilibrium Analysis,"
under Incomplete Information:
50, March, pp. 443-460.
Econometrica,
__ _ ___ __ _
i___
III
Milgrom, P.R. and D.J. Roberts.
Entry Deterrence," Journal
pp. 280-321.
1982b.
"Predation, Reputation and
of Economic Theory, Vol. 27,
"Dynamic Equilibrium Limit-Pricing in an
1983.
Saloner, G.
Uncertain Environment," M.I.T. Department of Economics working
paper.
"Entry, Capacity, Investment and Oligopolistic
1977.
Spence, A.M.
Pricing," Bell Journal of Economics, Autumn, pp. 534-544.
Spence, A.M.
Market,"
1979.
"Investment Strategy and Growth in a New
Bell Journal of Economics, 10, Spring, pp. 1-19.
Proceedings_ of the First ORSA/TIMS
1984.
Stecke, K., and R. Suri.
Ann Arbor,
Conference on Flexible Manufacturing_Systems.
Michigan.
Stigler, G.
Journal
Production and Distribution
1939.
of Political Economy, 47, 305-328.
Van de Panne, C.
Programming,
in the Short Run.
Methods for Linear and _uadratic
1975.
North-Holland, Amsterdam.