h
nD28
y
.M414
INS
ft^JUL
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A Stochastic Model of Resource Flexibility
Lode Li
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
02139
MA
April 1987
*1999S8
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
6
1988
A Stochastic Model of Resource Flexibility
Lode Li
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
MA
02139
April 1987
* 1999 -88
Abstract
We
study a stochastic control model of a two-stage production process with flexible work-
stations.
The cumulative productions
processes with
spectively.
An
random
intensities
of
both stages are governed by two Poisson counting
parameterized by production capacity and
optimal barrier policy
is
characterized by the critical values which indicate
when one
stage of production shoiild be turned
assist the
other station and
when
flexibility re-
to switch
off,
back to
when one
its
station should be switched to
regular task.
Both the conditions
for
optimal barriers and the expected total discoimted profit arc explicitly determined. Thiis,
the economic value of resource flexibility can be measured by the difference in expected
utility a firm
Key Words:
can optimally realize by possessing
it.
Flexible Resource; Poisson Process; Intensity Control; Barrier Policy.
'This work has benrRted considerably from discusnoiu with Erhxn CinJar, Mirharl Harrinon, iuid Morton Kami*n. The
author 19 greatly indebted for their many u.<eful stiggestions and commeiAs.
LIBRAWES
RECEIVED
1.
Formulation of the Model
recent advances of microprocessor-based manufacturing technologies, the
Accompanying with
manufacturing systems of the jnesent and near
flexible
reduce the changeovers between
futiu'c
products to a matter of seconds. The questions of how to manage the
to quantify the value of flexibility bee
ome
increasingly important.
resources affect not only the market competition
among
production processes. The economic value of the resoxirce
As a
by ignoring either of the two.
study the economic value of
in
expected
first
stej),
can oi)timally
The
j^resence of the flexible
not be acciu'ately captured
flexibility will
the paper develojis a stochastic control
reali7,e
and how
firms but also the internal control of
framework, that
flexibility in the decision theoretic
utility (profit) a firm
flexible resources
by possessing
is,
model
to
the difference
it.
In general, the economics of multiple-use resources can only be discussed in the context of
a
model that has uncertainty,
seasonality, or in general terms,
some
sort of variability.
It
also
is
necessary that one's model explicitly recognize the existence of more than one type of task or
processing requirement within the firm.
Thus, to consider any issue
economic value of
in the
A
resource flexibility, we must consider a multi-stage process (some sort of network structure).
simple model of two-stage, two-task production process
considered as follows.
is
Consider a production process that consists of two work stations,
an intermediate good
1
cau produce good
and
1
1
(^2/?
(0
^
1^2
^
good
2,
produce good
2
it
can produce good
using good
1
maximum
to the other.
In the context of service infhistry, one
is
2,
two tasks, producing
2 at rate (^ja (0
an average rate
at
/?.
rates,
and can
also switch either station frnjii
may
or produce
{B{t),t
>
0}
,
Z{t)
A,
B
good
is
the
amovmt
1
think of the model in terms of that
1)
and output (production
where A(t) and D(t) denote the amount of production
of production 2 in the time interval (0,t].
where x
1).
one type of production
represented by two increasing non-negative integer-valued stochastic processes
B=
<
i^i
completed only when two presetiucnced requirements are finished and the inventory
customer wait. The cumulative input (prochiction of good
and
<
Suppose the management monitoring the production rates within the
!)•
range of their
service
and
and an inventory storing intermediate goods. Station
an average rate a, or
at
Similarly, station 2 can
at rate
a final
1
Then
the inventory level at time
=
A(t)
x
+
of inventory at time zero.
are independent Poisson processes with
-
1
good
the
2) are
{A((),<
>
0}
and the amount
is
D(t).
We assume
random
t
A —
of
is
(1.1)
that
intensities {ai,t
>
O} and
{/9/,t
>
O}.
(1.2)
That
is,
{A{t)
-
J^a,ds,t
>
0} and {B(t)
-
J^,
ft,(ls,t
>
then be viewed as the actual production rates of i)roducts
0} are martingales, and a, and
1
and
2 respectively.
/?,
can
Given the capacities n
aiul
and the ineasnremont
/?,
zero), a feasible operating policy
of flexibility
(S,
and
(selected at time
/>2
defined as a pair of storhastir pif)resses (at,
is
ft,)
that jointly
satisfy the following:
(1.3)
(ai)
and
(/?,)
are left continuous
(1.4)
(ai)
and
(fti)
are
(1.5)
<
a,
for all
Z{t)
(l.G)
<a
> 0,
t
+
non-negative for
is
Condition
+
(«
<
Uo)(Pi) ^nd
hft)
ft,
<
l(o,„l('^^)
ft
+
+
(P
-^i")
'
l{o)(a(),
all (.
implies that q, and
(1.4)
limits,
adapted with respect to Z,
l(o,/9](/?/)
•
and have right-hand
control that the firm exercises at time
In condition (1.5), for any set S,
t
ft,
is
are functions of (Zia),.^
<
This says that the
t).
based on only the historical information before time
ls(>'') i? fui
^^"'
indicator function,
.1,
if
'0,
otherwise.
t.
i.e.,
.<.G.9;
For example,
<
if
«l(o,/?l(/?f)
Thus, condition
than waiting
+
(a
Z The
.
if its
(1.3),
the
/)
dollars.
demand
good
1
for final
assumed
(ft,)
of intermediate products can not be
fornuilation,
Bremaud
we
ft,
<
ft-
0.
of the workstations.
are integrable
good
2
and predictable
has no choice other
met from stock on hand. For the
(1981) and Li (1D8C).
specify the cost structure as follows.
Each
final
product
goods
is infinite.
and
The plant
C2 dollars
per
incius a linear variable cost, Ci dollars per unit of
luiit of
material cost and also labor cost
if
good
The
2 actually prochiced.
variable cost
workers are paid piece-rate. The selling price p
to cover the total variable cost of a final good, that
is
;>>ci-fC2.
A
physical holding cost of h dollars per unit time
held in inventory.
Assume
is
for
(1.7)
incurred for each unit of intermediate goods
that the firm earns interest at rate
on the funds which are required
infinite
is
simplify the market side where the firm serves as a supplier Ijy assuming that
actually produced
may comprise
is
We
and
=
flexil)ility
restriction (1.6) implies that the i)roduction of
demand
To complete our
if ft,
space parameterized by the
also ensures that (a,)
it
justification of the formulation, see
worth
'^2/?)l{o}(/?r)= {^"^,.
h^ft.
(1.5) reflects the strategy
Together with conditions
with respect to
+
r
>
0,
compoimdcd
production operations, and the production
time horizon. Therefore, given that the
initial
inventoiy
is i,
is
continuously,
planned over an
the expected profit
is
/•oo
w(x)
=
r-"\pdD(t)- cidA(t)-C2dD(t)- hZ(t)dt]},
E,{
Jo
(1.8)
.
=
where E^ denotes the expectation conditional on Z{0)
Applying integration by parts theorem to
we have
(1.8),
=
n(x)
x.
h
-
v(x)
-,
(1.9)
r
whe re
roc
= E^{
v(x)
-
e-''[qdB(t)
/
vnlA(t)]],
(1.10)
./o
=
q
—
p
—
h
C2
-\
and
,
_
=
w;
—
h
r
r
The problem
management
of the
the expected profit, eqnivalently
By
(2.3)-(2.G) are satisfied.
(Ill)
Ci H
to choose a pair of control processes {a,
is
ft)
to maximize
such that assumi)tifui (1.1)-(1.2), and feasibility constraints
t;(.r),
the Poisson assumption, the olijective function imder a feasible policy
can be further written as
=
v(x)
where q >
2.
w
eAJ
c-"(qft,-wa,)dt\,
(1.12)
following from assumption (1.7).
The Barrier
Policies
histead of solving the general intensity control problem set up in Section
barrier policy,
producing good
1,
we require that station
some
until inventory reaches
works
1
level
or switches to assist station 2 until the inventory has
2
it
works
on
at full capacity
inventory level
critical
regtilar task,
is b
numbers,
and resumes
down
rate) brings the inventory
at zero inventory level
is
its
either waits or switches to helj^ station
There are three
accumulated to a
we
namely, Imrricr policies, and establish some computational
class of feasible policies,
By
1,
h,
its
to
h^.
1
and
—
at this jxiiiit
li,
been
2.
The production
62-
when
it
liy
on
results.
its
^i
units; similarly station
until the inventory
of
regular task,
either ceases production
is
zero,
been increased by
good
1
is
the i)roduction of good 2
turned
off
good
of
1
then
62 units.
when
the
(maybe with higher
the other hand, the prodtiction of good 2
and resumes when the production
level of 62-
d(^i)l('f(Hl
until the inventory has
On
ftj.
cai)acity
full
its
producing good
regular rate
h
at
investigate a
is
ceased
(again at an increased rate)
Obviously, only on-off strategies are allowed in a barrier policy,
namely
a,e{0,a,a +
and the essence
process Z, Ei
=
generic barrier policy,
{b — hi,
.
.
/?,
G
{O, /?,/?
+
.^1
a},
becomes an optimal stopping problem.
of control
To characterize a
6ift),
.
,h},
and E2
=
\vt
{0, 1,
E=
.
.
.
,
{0,
^2}-
1,
.
.
.
,
To be
6} be the state space of the inventory
general,
we allow
cither
work station
switch back and forth between joining the other production process and staying resting when
its
regular prodviction process
=
with \Ka-\
1,
is
.
.
.
,
^iifl
J/tA-
— t/ii + l},
ft
turned
\^ik\
=
A'Ji
station
off,
is
{''
ceased.
~
ordered
''''pi'f'sciits
— J/ii
sits idcl
i
y'tk'
An
seciiieiire of disjoint sets,
a division of set Ej,
— Vu + 1}
''~!/u
=
i
<^'iven
<'*^-
K,i
/f,'j
,
,
.
.
.
,
if,,,,,
Kn =
1,2, e.g.,
production of good
,
/<",„,,
{h,b
—
^
j)
i [i
and station j works alone jnoduring good j when Z[t) G Kn;, and
=
=
=
both stations work together when Z[t)
G
choice of sxich a sequence represents
the possible order combinations of switching between using
station j (j
K'fc,
=
—
^
alone and using lioth stations.
i)
u'l'^iK'i^,
all
and
= Ei=i
j/,-
Va-. for
=
sinii)licity,
follows that "^^'^^
It
= M{t >
1, 2,
A;
=
1,
.
,
.
.
m,-
K
Q
^
and/or
denote
y'-^
=
h,-
if,',„^
/f,
-
=
j/,-.
0, the
Uji.J.j/f,/;.,
The value
turns out with the help of station
i)
i.
=
0,
t
^
T(T{f„-i,h),h
T„
= r(<„ft =
-
AT(T{f,.-i.0)J>2).
b,)
=
i?°„
r(r„-i.n).
5^;?""
fci),
=
r(<,.ft2),
1,2
barrier policy then can be written as, for
where No
e K},
r(T'„-i.'')AT(r„-i.O),
r(r„_,,t),
and n
S,Z{t)
E. Let
=
s:'„'^'
i -=
1.2.
fo
<=
A
For notation
/f,i
T[K) = m({t > 0,Z(0 e K),
any stopping time S and any subset
I
allowing
1. 2,
also define
T(S,K)
for
=
»'
t
represents the total imits that production j (j
t/,
We
for
if,V For
(
>
0,
a,
=
a{lr.'At)
+ 1n,(0) +
(«
+
'^2/?)1a-(().
(2.1)
ft,
=
ftilKAt)
+
(/^
+
f>i^)lx[it)-
(2.2)
= U^ o(r,., r,.+.],
TV.
= U^
1a',(<))
U';;^,,
^
+
(^.^
and
/?•.].
N'-
= U^,
U';^%
{RLs^^']
for
among
five
1,2.
In fact, the inventory process
Markov
Z under
a barrier policy
is
constructed by switching
processes: a difference between two Poisson processes with intensities
two negative Poisson processes with
with intensities a and a
+
intensities
ft
and
ft-Vfiin respectively,
n and
ft
respectively,
and two Poisson processes
S2P respectively.
Proposition 2.1. The va/ue
fuucfioii v{x)
under a barrier
j)oIiry deBiird in (2.1)
and
(2.2)
is
of
the form
v(x)
=
-(qft
-wa) +
i(x),
(2.3)
r
where
v(x)
=
avini(x)
+ a{w +
{>iq)u\(x)
—
ftqu2(x)
—
ft(q
+
62V')u'2(x),
(2.4)
-
TOO
(2.5)
Jo
roo
=
u'-(x)
E,{
e-''l:,.,{f)dt}.
(2.G)
Jo
Proof. Note that
+
1a'„(0
The propositiou
1a',(0
+
+
1a';(0
and
follows by pubstituting (2.1)
+ 1a-(0 =
1a',(«)
1-
(2.7)
(2.2) into (1.12). roi)larinR
1;\'„
using expression
and collecting terms.
(2.7)
Tims, the value function under a barrier policy obtains by computing
steps in the computation. First
Then, show that
and
u,
u-
compute the Laplace transforms
can be expressed
Denote the Laplace transform
and
u'-.
We
of the hitting times T{b)
take two
and r(0).
terms of them.
in
by
of a hitting time
=
0(x.y)
u,
-rT[y)
E,
Then
<
Proposition 2.2. For
x
<
h.
qib
—
e{x)
x]
(2.8)
f>)'
9(1')
where
—7
g(x)
=
(itPi
r.(x)
=
(/,^j
\
f
—J
*
-
a P2
,
(2.9)
pi
and
P2 are the
<
pi
<
I,
(C p2.
two roots of the qimdmtic equation
r
with
-
P2
>
I,
a.
=
d.
= i-
1
-
for r
>
a
=
.
ft
p'-^
a
+
ft
+
r
a +
'
+
ft
(2.10)
r
0,
a*
{(ip^'riri.p^')'"-",
=
1
- (^.P^'r
(riiP;')'"-"
(2.11)
U2P2r{n2r2)'"-'\
d'
= i-
U2Pir(n2Pi)'"-'\
and
P
-,
(i
ft
Proof.
We shall
+
r
only prove the
Define a function
i()
6=
first
Q +
f/i
ft
=
r
ft
+
+
h<^
fiia
+
12
n +
=
r
a +
dift
52ft
+
r
equation in (2.8) and the second follows from a similar argument.
by
i>(x,y.z]
_
Pi
-
P2
y—
pV
P2
for
x.y.z^
E.
(2.12)
—
By Lemma
3.3.2.1 in Li (1984),
we have
=
^:rle"'^'l|Z(r,)=n)]
where Ti
=
T{0)
A
and
i'{x,0.h).
iE.['~'^' l|z(T,)=ft|l
^
i'(x,h,0),
T(b) as defined above. Tims,
=
0(x,O)
(2.13)
i>(x,0,h)-\-i'(x,h.ri)0{h.O).
Particularly,
${b
-
=
6i,0)
i'{h
-hi,0,h)
+
rl>(b
-
bi,h,O)0(h,O).
(2.14)
Also note that
d(h,0)
=
(2.15)
ei'n\'~^'0(h-l>i.O).
Substituting (2.15) in (2.14) and using (2.12), we o1)tain
^(6,0)
el"r,J'-'"V'(''-6i,0,6)
=
i-eS"„J'-'"V'(6-ti,fc,o)
- {LP2'r(niP2')''-") - UiP2')'HmrV)'"~")p:'' -
(1
-
(1
(1
(1
{^^r:')"(>hP;')'-'-'")
(2.1G)
h
UiP7')"i^hp:')'"~'")P2
9(0)
9{by
The
first
equation
in (2.8) for
any x
^ E
follows
by substituting
(2.1G)
and
(2.12) into (2.13).
I
Note that the Laplace transforms of the hitting times T(0) and T(h) are independent of the
switching orders
when one production
process
times in Ey or E2 will not change as long as
is
t/i
turned
off
(|^'^i|)- J*"*^'
and Z
''i-
'
is
=
in i?j or i?2
1- 2,
because the sojourn
are fixed.
Proposition 2.3.
e(x,b)
Ai
,,
._
a;
0{x,b)
(2.17)
6(x,0)
X,
«2(i)
=
—
r
X',
:
•
..„
>,,-„,.,,
0(x,0)
-
^.
-
"2(2;)
''
i-e2^'^^-''^fl(62,0)'
'
r
l-e2''h'~'"0{b,,Oy
where
k=l
k=i
(2.18)
.
fori
=
1,2.
the definitioii of a banirr policy ((2.1) and (2.2)), for oacli n and k. /?j„
By
Proof.
sojourn time in the states, {l,
.
,
t/u}.
the sojourn time in the states, {l,
.
.
.
.
.
E[r-^^^l.-'i..^]
Also note that R^
n =
„_^i
1,2,..., are i.i.d.
—
^ Poisson process with intensity
/?,
t/u}i for a Poisson process with intensity
and E[r-'^''^ -"'"^]
= er-
=
and
/?
5^,, is
—
5/,,
+ i^ia.
the
i?j„ is
Therefore,
,/,"
the time between process Z's two consecutive entiy of the state
i?i.„,
random
,
ff""
—
h,
variables. Hence,
yoo
.
OO
^'
III
'"1
I
c
"•>•
= if;X;^.[l-e-^'<-^-']£;,[e-'<.e-^n=.'"'"-'^'"%-^^.'-'<^'"-^'^
*"
n=l *=1
-mi
c»
''
fi=l
A-=l
*"
i)=n
r
The
l-ef'r,;'-'"/?(6-ft,,6)'
rest of the proposition
can be similarly proved.
I
Propositions 2.1, 2.2, and 2.3 give us the explicit form of the value fmiction imder a barrier
policy.
3.
When
to
Nap and When
In a barrier policy,
station j {j
be,
=
is
Help
production of good
producing
its
assuming that the parameters
what
t"
j^ i) is
if
to
assigned
j/,-
h, bi, 1)2, j/i
the best order combination
among
t
stopjied, then station
is
r
can have a break while
units.
The question
f^ud
are fixed. In other words,
all
j/2
is
when
the possible choices,
these coffeebreaks should
we want
to determine
(/C,i, /<'|i, ..., K,„,,, /<",,„_),
1,2.
Let
mi = 2,^11 =0,K;i = {^.''-1
6-Vi +
l},/i:i2-
m2 = l,k2i = {0,L...,U2-l],k'2i =
{t/2
[h-Vi
h-h, +
i],k',2
=
^,
62-1}.
(3.1)
Proposition 3.1. For
choice (3.1)
is
fixnl
b,
rihI
hi,
=
i
)/,,
the })nniri policy with the switching order
1,2,
the domiiicint strategy.
Proof. Denote by A,
and
Aj,
=
t
i.e.,
(3.1),
-
Ai
-
(1
111
Ci
),
-
'^i
1
-
Notice that in the vahic fiuiction, the order choice, Kii,K[^,
only,
under the order combination
1,2, the functions defined in (2.18)
whereas the order choice,
suffices to
/<'2i, /C21,
.
.
.
,
,
.
,
affects the vahics of Ai
and
affects the vahies of A2
,
•
»?i
.
A'j
only.
and
Therefore,
^
Aj
it
prove
awXi + a{w +
(^i7)Aj
— awXi —
a(vi
+
(^i7)Aj
>
0,
(3.3)
+ P{q + hw]\\ - pq\2 -
ftqh
P{q
+
('>2W)^2
>
0-
Also note that for any order choice.
and
-
1
=
'/,
(1
z^'7,
-
ni
(3.5)
),
1=1
since X2^!'^j v.t
=:
j/,-
and Efc^i
v'a-
QwAi + a[w +
=
a'^i7('^'i
=
"'^17
-
=
''t
(*'i7)Aj
-
Hence,
!/••
— QwAi — a(w +
(^i7)Aj
-^1)
2^(1-^1
(l-r/i'*)>0,
)'?!
it=i
ft(q
+
PS2W{\\
—
ftqXi
=
+
^2«')'^2
>
and
62
>
/?7'^2
-
ftiq
+
('>2V')'>^'2
A2)
= Phw2_^i2
if (5i
-
(1-^2
^2
)(l-'/2
)>0'
0-
I
Suppose the management
feels that the
upstream station
in-process inventory reaches a upper limit and the
inventory
is
down
should
first
break when the work-
downstream station deserves a break when the
to zero. Proposition 3.1 suggests a "iirinciple of coffeebreak" for the
of the flexible workstations:
1)
fleserves a
switch to
when
lielji
the
management
the inventoiy hits the upper limit, the upstream station (station
downstream station
8
(station 2)
and then take a break; when the
empty, the downstream station should take a l)ieak
iuvciitoi7
is
upstream
station.
Proposition 3.1 rules out
many dominated
egy space in our search for the optimal barrier policy
4.
to help the
significantly reduces the strat-
in the following section.
The Optimal Barrier Policy
Using Proposition
values,
2
and henre
strategies
and then switch
first
/),
when
level
is
hi
and
j/,-,
=
i
1, 2.
That
is,
station
1
stops
the inventory level reaches the upper limit
down
to
—
&
at this time, station
+
/>!
2 starts a
the inventory
up
By
up
t/i
until station 2
takes on
1
hand, station
level further
can be characterized by
3.1, the set of barrier policies of interest
it
regular task and switches to help station
its
b,
station
works alone to bring inventory down to
regular task and the normal production resumes.
break right away when the inventory
-
2.3
and
we can write the value function
(3.2),
b
On
—
hi,
and
the other
level hits zero until station 1 brings
1
and then switches back to the production of good
Propositions 2.1
when the inventory
starts a break
1
to y2, at this point, station 2 switches to help station
to b2.
five critical
bringing the inventory
regular task).
2 (its
dominant barrier
luider a
policy in terms of the five critical values.
awnl^-'^{i
-
v{x)
ei")
+ a(w +
Siq)(l
-
r,!'-"'
„''2-!'2\„
y^^ _L /9^^ _L X ...^fViM
,Lr\
-
d'
a*il,p\
—
atd*P2
)
where
a. (61,1/1), a*(6i.t/i). ''(''21
Proposition 4.1.
optimal policy must
Proof.
(x
>
1/2). ^ik^I f'*(''2,J/2) ^i'''
In an optimal bHiricr policy,
—a
Let
T
be the
first
policy,
time the inventory
reaches x again after T, and ii(x
defined in (2.11).
=
0-
fti
=
ft
in certain state
by one unit, 5
=
T(T,x), the state
mii.sf iiave
—
1,
x)
we have
rti
level decreases
be the expected
following the optimal policy over the i)criod (T, 5].
of the process,
By
=
i/i
</2
=
0,
and
i)resent value starting in state x
the strong
—
C2
—
xh/ft
—
1
x
and
Markov and renewal properties
x/,.
p
an
^" other words,
we have
"^^^=l-ii;,le--]^,_i[e--]<''-^^-
Note that v{x)
dtp2
fully utilize the flexible resources.
Suppose imder an optimal barrier
0).
it
(4.1)
^•„~(*~^)
„-(*-•')
a^d py
J
p'.
>
if
+
u(x
and only
—
1,1)
>
if
p
-
C2
—
xh/ft
since there
is
+
u(x
EAe-r^
-
1,
z)
>
0.
+ u(x -
1,
.))
There must be the case that
a barrier policy that can guarantee zero profit.
,
f^i
=
+
("ijQ.
Denote by
v(^')
the value function under the alternative policy.
^>'(x)
=
!!^
_,si
,
since
>
r?i
^1
—
and p
—
C2
+
xh/(fl
contradicts that the nominal policy
= P
that Pi
By
-\-
We
(^i«-
where
whicli follows the optimal policy cxcejjt iu the state x
Now, choose an alternative policy
f>ia)
-
(?'
+
-
T—7- +
-
l,x)
^2
u(x
>
u{x- l.x))>
-
]>
cj
-
=
can similarly prove that Pi
implies that ni
v(x)
+ u(x —
xli/fl
optimal. Hence, in an optimal barrier
is
Then
1,
aj
i)olicy,
= a+
x)
=
>
0.
This
implies
(^2/^-
Propositions 4.1, we can further reduce the set of \uidominated strategies to be the barrier
numbers
policies with three critical
dominant barrier
A,-,
i
=
and
1,2,
by assuming
h
;/,
=
0,
=
i
1,2.
Under a
policy, the value fimction
a(w +
=
v{x)
{>iq)(l
(Lpl-d'p2
-m')
—
a*dtp\
a^d* P2
(4.2)
p[q
+
62W){\
-
iup-,^'-'^
r,^')
a^d
r
pi
-
a>2~"~''
—
a dtp^
where
=
a.(/>i)
1
-
{niP2')'','i'(l'i)
=
1
-
(niPl')''
=
'^(''2)
,
1
-
{^l2P2)'\d'(h2)
=
1
-
(fhPi)'"-
(4.3)
Define
=
l(h-hiJ,2]
By
l(b
Fact
—
bi
in Apjiendix,
5.
+
1;1,62)
is
Thus, we can define
we know that
^2
any
n'(b^)d4h2)p\.
b,l)iJi2
and
l(b
—
b
>
i;''i-l)
'f'
such that
/*2
+
(4.4)
bi
A
b2,
l{b;bi,b2)
>
0,
strictly decreasing in 62-
be such that
l(h
and
for
strictly decreasing iu 61,
bi
-
a4b,)d'(b2)pl
-
ii
+
-
62)
>
0,
and
b^- 1, /<2)
<
0,
(4.5)
l;/'i,l)
>
0,
and/(/)-?/i;/*i,l)
<
0.
(4.G)
1; 1,
l(b
be such that
l(b-b2 +
Lemma
4.1. For any x
and
there
an optimal harrier
fixed
h,
62.
ma.x{v{x;bl*),v(x,b)} where v(x;y)
is
the value function with
is
by
—
«/
ftj
such that v{x;b\)
'i"*^
''J*
's
=
uniquely de-
termined by
A:i(^i
+
1)
<
!indki{b^)>
r~'
10
—
(4.7)
«1
=
''l
.
1
ft(l-n2')n(ln)
ami
n{hi)
=
m,(6i)
=
(u(fti
-
1)(1
-
-
a,(b:)(l
-
r?J'"M,
m'{b,)
=
a'(b,
-
1)(1
-n\']-
a'(b,){l
-
ryj'-').
ki is strictly decieHsinp; in bi for b^
Proof. For fixed
and
b
—
nt(lii)a*(bi
<
a.C'i
r/t')
and
bi
-
1)
-
l)a*(fti),
inrrrHsing in
is
(4.8)
bi for b^
>
bi.
compute
^2,
v{x;b,)
-
v{x;b,
-
'L^)
= K, (kAbi) -
I)
where
rl(b;b„b2)l{b;b,-l,b2,)
since a(bi)
>
<
e(x)
0,
>
(defined as in (2.9)), and l(b\bi,b2)
by Facts
and
4.,
2.,
5.
in
Appendix.
We
also notice that
,,;,,,,
- ki(bi
-
ki(bi)
,,
1
= «^f
'"^(1
-
r„),r'^-^-^^V-""''^''a.(l)a'(l)^(/'
-
f'i
+
1;
1.^2)
7
/?(l-r7h'i(''i)«(''i-l)
li;
•
- Pr)^'-'-" -
n';((i
(1
-
p:')P2'-'~"
-
iPl'
-
P2'))]
<
»l=:0
if
(1
and only
—
py
)p2
if
>
decreasing for
-
l{b
61
Pi
<
+
l;l,/'2)
>
~
p2
fo''
>
hi
6J* (defined in (4.7))
LGmma
b^
and
and
is
h
a;
>
since a.(l)
])y
1
Facts
1.,
>
hi,
increasing for
bi
0,
a*(l)
and
3.,
<
4.
0. a(bi)
>
and
0,
-
(1
p2^)Pi
Therefore, ki{bi)
in Api)cndix.
and there are two possible
is
local optimal points,
(the corner solution).
4.2. For any x and Rxcd
b,
ftj,
n\SLx{v(x,b2*),v(x,l)} where v(x\y)
is
the va/iie function with 62
there
is
an optimal
})Hrrier
=
V-
ftj
-'^''c'j
^'J'^
''2*
f^iat
''*
v(i;
frj)
=
imiquely de-
termined by
k2{f>l
+
l)<^—-^,
w+
andk,(b;)>^—-^,
w +
Oiq
(4.9)
OiQ
where
k(b)=
'^ '^
a{l-r,\')d{b2)
mn'(b2)a4bi)p'2
-
n4b2)a'(bi)p\y
d{b2)
=
d.(b2
-
iyr(b2)
n(b2)
=
d'(b2
-
1)(1
-
r;*')
-
n,(b2)
=
d,(b2
-
1)(1
-
r,5')
-
11
-
-
1),
d'(b2)(l
-
r,*^-^,
d4b2){l
-
4-'-'),
d4b2)d'{b2
(4 10)
and
k2
strictly increasing in
i!^
PtooJ. For fixed
b
and
is
-
>
We
strictly increasing fnr
=
>
l>2
''2
o
+
('low
\
J-
.
Vi
ApM'^ + hq)P{n[b2)a»{b,)p\ -
n,{b2)a*(b,)p\)c,{x)
^
^
rl[b,b,M)l[h\hi.b2-h)
^2)
>
n*(b2)a.{h)P2
"
(defined as in (2.9)),
due to the facts a. >
is
/
-l) = K2[ hib^) -
v{x- b2
the vahie fmictioii with ^2
^
^'"
since g[x)
nnd
b^
compute
61,
v{x; b^)
where v(x\y)
<
for h^
1)2
<
0, a*
0,
l{b;
ftj,
>
n'(b2)
and
0,
"*(''2)'i*(''i)/'i
and n,(b2) >
>
hy Facts
and
2., 3.,
4.
in
Api)en(hx.
have
also
- r72)(l - ^?^ )^f '-'^^^-\/.(l)'/'(lK(fc " h + 1;
ft({n'(b2}<u{b,)4 - n.(b2)a'{by)p\)((n'{b2 - l)a.(b,)r\ - n.(b2 aril'-'-'d
'^
''
'^
'
'
•
lE^2(^2 -
-
/'I
({P2
- Dp;'"'-'-" -
-
(P:
1)/'2'"'"'"'")1
^'1,
1)
l)a*(''i)/i)
>
11=0
if
and only
[pi
~
l)/'2
We
if
^(6-
+
^"''
'
'
61
1,62)
1;
>
2;
1
>
<
since (1,(1)
by Facts
1., 3.,
and
>
0, fi*(l)
4.
^udp2-Pi <
(/'2
-
l)/'r'*'"^""'
"
Appendix.
in
conclude the proof by the same argument as
0,
the i)roof of
in
Lemma
4.1.
I
Lemma
and
4.3. For any x
fixed
6,, t
+
>
)k(6*
1)
=
1,2, fhcre
+
^—
^-.
a
;.s
iwique optimal harrier
bou)
w +
+
7
^
andk(b')<-
SiQ
(jJ
b*
determined by
i^2W
(4.11)
f-,
+ Oiq
where
-
"(1
-
n\')(aAf'iK{l>2)(i
-
P2')p;''
-
n'(fn)<L(b2){i
-
ft(l-n'2')a,(bi)a*(h,)(r2'-p-[')
and k
Proof.
is
It
a strictly increasing function of
ft.
can be calculate that
v{x;
where v(x\y)
is
ft)
-
/q
v[x-
defuied in (4.2) with
j^
^
h-l) = K\
-\-
\w +
ft
=
pIpI^x^uj
—-
S-)UJ
k(b]
oiq
j/,
+
6,q)ft,u(hiy('>l)(P2'
r/(ft;fti,ft2)/(ft- l;fti,ft2)
12
-
pT')
>
Q
p:')p2'')
(4^0)
since a*
>
0, a*
<
0, e(x)
<
and
pi
<
P2-
I
Proposition 4.2. There
which jointly
exists an
satisfy conditions in
optimal barrier policy with three
Lemma
Proof. Given the monotonirity projierties
determined by (4.11)
is
critical
numbers
h'
,
6,*, i
—
1, 2,
4.1.-4.3.
shown
bounded inflependent
in
Lenuiin 4.1
of the choire of
-
4.3,
/*,,
it
ran he shown that
/**(/'!, ''2)
and hence, there always a fixed
point sncli that conditions in Leiuiiia 4.1.-4.3. are satisfied.
I
5.
Conclusion
In this paper,
flexible resources.
we propose
a stochastic control
With respect
model
to study the managerial issues regarding
to a class of feasible policies,
namely barrier
policies,
we present
general computational results, identify the dominating strategies in the real time allocation of the
flexible
workstations which can serve as a i)riuciple in the
management
of the flexible production
system, and find the conditions which explicitly determine the oi)timal barrier policy. The importance of the barrier policies
lies
on the
fact that the
optimal barrier policy
is
an ojjtimal Markovian
policy in the sense that the state space consists of inventory and workstation status. However, the
completeness of the optimal barrier
jiolicy
(whether
needs further investigation.
13
it
is
optimal
among
all
the adai)ted policies)
.
Appendix
We
Lemma
list
the following facts as
refcrcucc of Section
tlic
and most of the proofs are immediate.
4.
5.1.
a^l)
=
-1
-
I
=
t]iP2
— (l-/'2
T—
+ (^lO + r
^(^2
+
-1
i^l)
-^
)
>
f''
)
<
0.
/7
a (1)
=
1
-
tup^
rf^l)
=
1
-
f?2^2
(/
(1)
=
——
—
=
r
-
1
+
r
a + O2P +
r
=
ri2Pi
n
a + O2P
,
=
—-(1 ,
r—
Pi
(1
-
/'2)
<
0.
(1
-
^1)
>
0.
Lemma
5.2. a«() anci d'{) arc strictly incvcHsing hikI (i*() niul '/*() nrc strictly decreasing.
Lemma
5.3. a(bi),'m.t{hi),m' (hi), d(h2),
i
=
1,2.
Consequently, a,(bi)/(i*
and d'(b2)/d,(b2),
(1
-
11,(1)2)
—
(hi), (1
r?i'
-
r?*')/(/.(f>2) a^'i (1
3«(i ""(''2) ^t-o positive
)/a,(/*i) an*/ (1
f;2')/rf*(''2)
a''^^
—
and increasing
=
a,(bi)a*(hi
-
-
1)
a^bi
-
nirrea.sinff Hi
/<2.
l)'i'(/'i)
6,-2
=
a.(l)n (I) 2_^(r]iPi
r?i'
p^
)
-
(p2
Pi
)
11=0
m,(6i)
=
a.(6i
-
1)(1
-
r/J')
-
a.(6,)(l
-
ti\'-')
m*(6i)
=
a*(bi
-
1)(1
-
r,J')
-
a*(bi)(l
-
n^'-')
A,
-
.
-2
>-^(i - r,iK(i) Ei^i^rM-d „
,
1
^
,
/
.
>
^r'''-'-'")
>
0,
11=0
d(b2)
=
d,(h2
-
-
l)d*(b2)
d,(b2)d'(b2
-
1)
*2-2
n*(62)
=
n'r'd.(i)d'(i) j2i'i2PiP2)"(p'r'-"
n=0
=
d'(b2
-
1)(1
-
r,^')
-
d'(b2)(l
-
p\'''-") >
- r;^-M
*2-2
n.(62)
=
ri2'~'(i
=
rf.(ft2
-
-
J2in2Pi)"{i - p'r'-") >
ri2)d'(l)
1)(1
-
r,*')
-
rf,(62)(l
63-2
ii=n
14
-
r,*^"')
>
r7i')/a*(6i) aie jncreasing^ in
Pror)/.
a(fti)
for h{
0,
0,
>
0,
1,
fti,
Lemma
5.4. g(x)
Lemma
5.5. For any
decreasing in
Proof.
bi,
>
and
0,
b,
l(b
By Proposition
<
c(x)
0,
g(x)
increasing mid r(x)
is
61,^2 sncii that
—
+
/)2
1; ''! 1)
>
b
'''
A
bi
62,
'('';
is (lrcrr;isin<r
^i. ''2)
>
0, l(b
-
in x.
b^
+
1; 1, 62) is
strictly decreasing in b^.
2.3.,
u\[x)
=
E,{r e-''\^-.{t)dt)
Jo
_
1
-
d4b,)p',-d'(b2)pl
r??'
r
'
-
a'(bi)d,{b2)p\-a4bi)d'(b2)p\
'h
pi
/(ft;ii,/'2)
The second
>
since e(x)
>0,
l(b\bi,b2)
r
which implies
<^{^)
P2
<
0.
assertion follows from that
l{b-bi + l:lb2)-l{l>-l>i\l,f>2)
=
The
a4l)d'(b2){l
-
P2)p''2~'"^'
-
third assertion can be similarly proved.
a*(l)./.(''2)(l
-
Pi)p\''"^'
<
0.
strictly
References
Processes nnd Queues, Springer- VorhiR,
Now
1.
Bremaiid,
2.
Ciular, E., Introduction to Stochastic Processes, Pioiitiro-H;ill,
P., Poiijf
York, 1981.
Euglcwood
Cliffs,
New
Jersey,
1975.
3. ('iiilar, E.,
"QtU'ucs with Arrival? and Sorviros HaviuR
Random
Intensities", niiiiio. Northwest-
ern University, 1982.
4.
Dynkin, E.B. and A. A. Yiislikevich, Controlled Markov Prores.ses, Springer- Verlag, 1979.
5.
Harrison, J.M., Brownian Motion and Buffered Flow, John Wiley
6.
Heyman, D.P. and M.J.
Processes and Operating Characteristics
7.
Heyman, D.P. and M.J.
Models
Sobel, Stochastic
Sobel, Stochastic
,
in
&
Sous,
New
York, 1985.
Operations Research, Volume
I:
Stochastic
II:
Stochastic
McGraw-Hill Book Company, 1982.
Models
in Ojierations
Research. Volume
Optimization, McGraw-Hill Book Company, 1984.
8. Li, L.,
A
9. Li, L.,
"A Stochastic Theory
Stochastic Theory of the Firm, doctoral dissertation. Northwestern University, 1984.
10. Ross, S.M.,
of the Firm", Forthcoming,
Mathematics of Operations Research.
Applied Prohahili.^tic Models with Optimization Ajiplirntions, Holden-Day, San
Francisco, 1970.
11.
Prabhu, N.U., Queues and Inventories:
Wiley
&
A
Study of Their Basic Stochastic Processes, John
Sons, 19G5.
"SbOk
U50
16
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